Obviously the techniques are different, but you're both solving a "problem" with the help of a computer.
What problem writing an amusing simulation of space ships shooting missiles is solving in your opinion? Other than the lack of such simulation, of course. Could you solve such a problem without a computer? I don't think many people would be amused flying a real space ship and shooting missiles.
Every solution involving software has many many layers of software inside it, and every single one of those layers is solving a problem of some sort.
Some layer somewhere in that expenses report is solving the problem of how to save and load data, for example. It has nothing to do with the business problem being solved, but it's still a technical problem that needs to be solved.
It's the same for games.
If you're doing a business application for a small shop (payroll system, cash registry, invoices printer…), good accounting is probably more precious than good programming: you have plenty of computing resources to waste, your programs can be relatively small…
But when performance starts to be key (video encoders, AAA games, databases…), the effort devoted to programming becomes comparatively bigger.
Search at Google, scaling at Twitter, any chip that powers your car's airbag, ...
So game programming is not a different animal, it's just one of the domains where performance is a lot more relevant than for more mundane domains.
The truth is game code often needs to be performant first, and the pursuit of that goal usually comes at the cost of maintainability and readability (and programmers' sanity), so I think further discussion is warranted.
This is true, of course. All else in this discussion is semantics, probably :)