When it comes to an inarticulate noise made by a person the two are quite interchangeable - although saying moan instead of groan can imply you are less sympathetic to the person making the noise. It can be said that inanimate objects groan (eg trees groaning in the wind) but not moan. There is a second meaning for moan: a person constantly complaining about something, especially a trivial matter, can be said to moan. This is not said as a complement.