Are you shooting film or digital - and if film, is it slide or negative (print) film? When you say that it "the camera doesn't capture it like you see it", that probably is the case and this is true in a lot of situations. Photography doesn't see the way your eye does, and that's part of the challenge and inspiration to making unique images.

The moon is much brighter than the clouds around it and really any form of photographic media (digital, film, etc) will not be able to handle that amount of contrast anywhere near as well as your eye will. Obviously it won't work in this case, but in other situations where there's too much contrast, try using fill flash to even out the light levels.