If you look at what LEO has to offer, you'll see the synonymous meanings as well as where the words differ. I think that should give you a good idea.

You can also look at the bottom of this page: http://www.oxfordadvancedlearnersdictionary.c...

Usually you concede a point in an argument or a business deal, something that's not personal but part of a larger concept. When you concede it, you give it to the opponent, you allow them to have it.

You admit a personal action, like a mistake or something you failed to do, something you feel guilty about, some knowledge you lack, etc. When you admit it, you confess that it's your responsibility and you ought to know or do it, but you don't/haven't.

I can't promise that that will work in every case, but maybe it will help as a general idea.