I usually have a fair idea of what key a piece is in. The pitches C and F are firmly ingrained in my brain, so I can often work out whether a piece is in either of these two keys. Occasionally, I get either of these keys mixed up with the key of Bb, as to my ears, these three keys sound quite similar and have a similar sort of tonal quality to them. G is one of my favourite keys, and I can usually hear that a piece is in that key. I usually sing the first line of "Jesu Joy of Man's Desiring" to confirm it. D is similar to G, A is similar to E, but all other keys for me have distinct characteristics. B is a very interesting key, because although it has one more sharp than E, it sounds nothing like it.
By the way, all this is personal opinion. And the keys I'm referring to are all major, since their relative minors have a similar tone, apart from the dominant chord.
As for improving your piano playing, I'm not too sure about that, as I don't think I'm a better pianist for it. Maybe as a composer or a pop pianist. Then again, being able to identify keys would probably aid the inner hear and help to improve aural memory.