As sounds, there is sometimes a difference. Someone in pain can both groan or moan, and the meaning is similar. However, in certain... umm, shall we say circumstances ;)... a person can moan with pleasure. 'Groan' isn't usually used in that way. Or let's say that someone tells a really terrible corny joke. The people listening would react by groaning, but not by moaning. 'Moan' can also be verbal, meaning to complain, as in "Oh, be quiet. You've been moaning about the bad service all day. I'm tired of listening to you". Groan does not have this meaning.

Basically they are synonyms with very little, if any, difference.