Video game development has several constraints that aren't commonly found in other development such as web development. In particular, for real-time games running at 30 or 60fps (frames-per-second), you have only 33 or 16ms respectively to process an entire frame of updates. Most game development targets fixed hardware such as consoles (PlayStation, Xbox) or mobiles (iPhone, Android), so if your code is slow, you can't just put a bigger processor in or distribute the code over an array of servers.
These are other requirements mean that most high level AAA games have to be written to be very highly performant, and in particular, to have reliably consistent latencies. Traditional Garbage Collected languages (Java, C#), whilst achieving high throughput, often struggle with predictable latencies as when GC kicks in, particularly in Stop-The-World collectors, you can get a 10-20ms pause which means you will drop a frame or two. This leads to stuttering which is undesirable. Even modern generational collectors still struggle not to have occasional bad pauses. If something like Azul's continuously compacting collector becomes viable for games, this might be solved, but until it does it is hard (but not impossible) to write high performance (soft) real time games in a GCed language.
The result is that most games are written in C++ (sometimes with a small embedded scripting language, mainly Lua, which uses GC) where memory management can be controlled. The downside of this is developer time - it takes large teams of developers lots of time to write games, because the code is largely written at a low level.
Rust is higher level than C++ and allows many useful and safe idioms, including a more functional style and better use of immutability. In general, like other higher level languages, Rust would allow a developer to be more productive than in C++, allowing games to be developed more quickly, with fewer developers. It does this whilst still allowing manual control over memory - some parts can be handed over to GC, whilst others can be carefully allocated to the stack or heap and managed manually. This allows predictable allocation and cleanup overhead, and therefore more deterministic frame times.
As a games developer, I am extremely interested in Rust as a potential game development language.
Disclaimer: I haven't actually written any Rust yet, only looked at code and thought about the potential. I've spent 5 years doing AAA console development in C++.