Programmers working in different styles will care deeply about entirely different aspects of programming. That's why you often see programmers vehemently debating the merits of opposing programming methods without ever reaching agreement. From each of their perspectives, their way is right. The defining difference in the argument is actually one of context.
To get a general overview of the different programming styles out there, and to get a better sense of what other programmers are concerned about with their code, let's look at the various programming styles. Some of these styles I know quite well because they are my day job, some of them I'm learning, some of them I've dabbled in, and some of them I know only from what I've heard. I'll try to be careful in what I say about those styles that I don't know too much about. There may be other styles that I haven't thought of, but these should be the main ones.
Embedded Programming
The main programming style I've been doing for the last few years is embedded programming. In this style, you are programming for a particular microprocessor or closely related family of microprocessors. The software you write is called 'firmware' and it will be programmed directly into flash on the processor or an external flash chip connected to the processor. Normally, this firmware is not changed very often, if at all, during the life of the product in which it resides. It's firm, not soft, hence the name.
Even within embedded programming, the differences in constraints from one programmer to another can be quite large, depending on whether the programmer is programming for a beefy quad-core GHz ARM processor or a meager 8-bit micro-controller. In most cases, the processor is chosen to have just enough horsepower to do the job it's meant to do, or more likely, slightly less power than the job requires, and the programmer has to make up the difference with crafty coding skills. Low power is a common consideration when choosing an embedded processor, and much of firmware design involves figuring out how often, how long, and how deeply you can put the processor to sleep.
Embedded processors have a number of communication interfaces and peripherals, and the embedded programmer must become well versed in bit-banging registers in the processor to configure these peripherals to interface with the outside world through attached sensors, storage devices, network interfaces, and user interfaces. Programs are mainly interrupt driven with real-time deadlines to meet. An RTOS (Real-Time Operating System) will provide mechanisms for defining interrupt service routines, tasks to run when interrupts aren't being handled, events and message queues for communicating between interrupts and tasks, and locks and semaphores for synchronization.
Programming environments are most often proprietary and provided by the microprocessor vendor. All of the ones I've used have been Eclipse based, and they provide a debugging interface with a hardware emulator that you connect to the processor to provide the normal debugging features of breaking into and stepping through code, reading and writing to memory, and giving access to the processor's registers. They also usually display peripheral register values in a decoded, human-readable way, show various characteristics of the RTOS state, and allow some level of profiling of running code. Non-proprietary IDEs are available as well, but they tend to be expensive.
In general, embedded programming happens very close to the metal, and you have to exert fine control over the resources available in the processor. Programming is usually done in C or C++, although Lua, JavaScript, and Python are starting to make inroads. It's as close to classic programming as you can get, with all of the considerations of limited memory spaces, special hardware instructions, and printing characters through a UART terminal included.
Systems Programming
Systems programming is the design and implementation of software that interfaces between hardware and other types of software, otherwise known as operating systems. Windows, Linux, iOS, Android, and the RTOS that an embedded programmer uses are all examples of operating systems. Systems programming is similar to embedded programming in many ways because the programmer needs intimate knowledge of the hardware, but whereas an embedded program normally targets a specific microprocessor, an operating system will run on a wider variety of hardware and include drivers for many, many more peripherals.
Operating systems provide the basic services that other programs use, like disk and file management, virtual memory, preemptive multitasking, device drivers, and application loading to name a few. The systems programmer has to worry about designing algorithms that will have high performance on a wide variety of hardware, writing drivers that work with an incredible variety of peripherals, and making sure that programs can play nice together without stepping on each others toes or bringing down the operating system altogether. Security and reliability are constant concerns of the systems programmer.
Most systems programming will involve at least some C and more likely a lot of C. The core of the operating system, referred to as the kernel, is normally written in C. C++ and Java are also commonly used outside of the kernel. The development environment is as varied as the programmer that's doing systems programming, but there are often a lot of specialized tools written specifically to support developers working on an operating system. Systems programming requires strong knowledge of algorithms and data structures, a meticulous eye for detail, and an ability to think about software running at the low level of the hardware-software interface.
Language and Compiler Design
Designing languages and writing compilers and interpreters is similar to systems programming in that programming languages are an interface between hardware and the software programs that run on that hardware. When the language runs on a virtual machine (VM) it even further blurs the line between language design and systems programming because the VM provides many of the same services as an OS. A programming language is not an OS, though. It's a lower level construct than an OS, and while an OS provides abstractions for various hardware features and peripherals of the processor, programming languages provide abstractions for the machine instructions and computational model of the processor.
Like embedded programmers and systems programmers, compiler writers are concerned with low-level performance, but they are concerned with performance in at least three languages—the host language that they're writing the compiler in, the source language that is being compiled, and the target language that the compiler outputs. After all, the compiler's basic function is to translate from a source programming language that other programmers write in to a target language that either the VM or the processor uses. Most compilers are written in C, although some are written in other languages. Rubinius is a Ruby interpreter written in Ruby, for example.
Compiler writers need to know the deep, dark corners of these languages to make the host and target code as fast as possible and to cover all of the things that any programmer could possibly do in the source language. Programmers want fast compile times, good interpreter performance, and the best runtime performance they can get, so compiler writers need to be well-versed in all of the low-level optimizations they can do to squeeze every last bit of performance out of their code, both the code the compiler runs on and the code it produces. On top of that, the code for the source language has to be interpreted correctly in all cases, even cases that most programmers will never think to exercise. Compiler writers need to think about and deal with complicated edge cases that result from interactions between seemingly simple language features that would make anyone's head hurt. I can't even begin to imagine how hard it must be to write a compiler for the more complicated languages (I'm looking at you, C++).
Application Development
The field of application development is as wide and varied as embedded programming. It ranges from the creation of simple utility apps like Notepad or a calculator to huge, complex apps like Adobe Photoshop or Firefox. The bigger an app gets, the more its development looks like systems programming, but at its core, an app provides a user interface and features to help the user do something useful with the computer. A computer with only an OS doesn't do anything meaningful. It's the applications that give the user the power to create and consume in all of the ways we've come to depend on.
For the most part, performance doesn't matter as much in application development. Normally an application has huge amounts of memory, storage, and processor performance to work with, and it's normally waiting for user input—a slow process compared with the high performance of the underlying hardware. More computationally or I/O intensive parts of an app may need to be optimized for performance, but the code should be profiled for hot spots and bottlenecks before blindly trying to improve slow operations. It's easy to get fooled into optimizing parts of the app that don't need it and ignoring the parts that are actually causing poor performance. However, in most cases the application programmer can get by with writing clean, clear code in a straightforward way and letting the compiler, OS, and hardware take care of making the app fast.
The application programmer has enough to worry about with making the app as easy to use and useful as possible. Choosing the right features, organizing the app so that it makes sense to most users, and creating an awesome user experience is insanely hard to get right. Users are fickle and demanding (not on purpose as they have their own lives and stuff they want to get done), and striking the right balance of features in any application is a constant struggle.
The environments for application development are generally quite good. Debugging features are much better than those for embedded programming, and IDEs like Visual Studio and Xcode make building a GUI and setting up the boilerplate code for an application much faster than it used to be. Application development can be done in almost any language, but native applications are most commonly written in C#, Java, Objective-C, and, to a lesser extent these days, C++.
Mobile Development
I'm not convinced that mobile development is much different than application development, except that it's done on a different platform than the target platform of the app. That gives mobile development some characteristics of embedded development because you need to connect to the system that the code is running on to debug it. Mobile development tools include a simulator to run the code on the host platform so that more debugging information is available while you're getting the app up and running. Otherwise, mobile development is very similar to application development in most respects.
Web Development
Web development is similar to application development in that the programmer is concerned with making a great user experience and producing a product that meets the needs of the majority of users for a given set of tasks. Also like application development, as a web application gets larger and more complex, development starts to look more and more like systems programming. The server infrastructure and services provided through APIs become the overarching concern as a web application grows and attracts more users.
Web development is different from application development in that network and database interfaces are a prerequisite for web development. A native application may have these things, especially the bigger applications, but a web application without them would be meaningless. The client and the server are linked, but undeniably separated, and the server needs to manage connections to hundreds, thousands, or even millions of clients whereas most native applications deal with a handful of users at most.
Web developers work in a wide variety of environments and languages. Text editors like Vim, Emacs, and Sublime are commonly used, but full-fledged IDEs like Eclipse, NetBeans, Visual Studio, or anything from JetBrains are also popular. To avoid reinventing a bunch of wheels in web programming, you'll use some kind of web framework, and there are numerous options in any language you choose. The most well known frameworks are (in no particular order) Ruby on Rails for Ruby, Django for Python, ASP.NET MVC for C#, Grails for Java, Laravel and Symphony2 for PHP, and AngularJS and Ember.js for JavaScript. Tools for web development are evolving quickly, and debugging requires a wide variety of skills from classic print statements and reading logs to modern site monitoring and testing with tools provided by various companies in the cloud.
Game Programming
I don't know much about game programming, since I've never made any serious video games, but I'll give this description my best shot. Game programming has elements of application development because of the game's user interface, systems programming because a game will normally have its own memory and storage management, and embedded programming because the game code needs to be as close to the hardware as possible and has real-time deadlines. The video game programmer is a slave to the frame clock, and everything that needs to be done for a particular frame—user input, computer AI, physics calculations, 3D model calculations, and graphics redrawing—needs to be done before the next frame gets drawn on the screen. That sequence needs to happen at least 60 times per second to make a smooth user experience.
Game programming may have more similarities with the next style, scientific computing, than anything else. The graphics programming is especially computationally intensive, and the graphics card is a massively parallel processor that's designed specifically for doing complex vector calculations. The most common game programming languages are chosen for their raw speed and closeness to the physical hardware that they're running on. That means C, C++, and assembly for most video games. Tools and environments are mostly custom and proprietary because they need to be so specialized for the type of development they're used for.
Scientific Computing
Surprisingly, scientific computing is most similar to game programming in that it is defined by massively parallel computation. While games model a fictional universe, the goal of scientific computing is to model a real-world system with enough accuracy to explore new truths about the system being modeled. Some examples of scientific computing include weather simulations, galaxy formation, fluid dynamics, and chemical reactions. Strong mathematics and algorithm knowledge is a definite requirement for scientific computing, and parallel programming is obviously common both when programming on a graphics card with CUDA and on a server cluster.
I'm hesitant to say that any of these programming styles is more complex or requires more skill than any of the others. Each style has its own set of issues, things that are challenging, and things that are interesting. They each overlap in interesting ways with other styles. Exceptional programmers populate every one of these programming categories, and what differentiates the programming styles is not how hard they are, but the types of problems that need to be overcome to create awesome software.