'Drunk' can be used as a noun or an adjective. For example, "He was a drunk." (used as a noun), or "He was too drunk to drive." (used as an adjective).
'Drunken' on the other hand, is only an adjective. For example, "He was a drunken mess." You can't call someone "a drunken", so that's where they differ grammatically, I think.
In your example, "a drunk/drunken man", both the 'drunk' and 'drunken' are used as adjectives, so there's no real difference beyond preference, but using 'drunken' is probably a bit stronger as an adjective than 'drunk'.
Also, I think people more commonly prefer to use 'drunk' as an adjective instead of 'drunken'.
Another noun you could have instead of calling someone 'a drunk', is 'a drunkard', but that seems a bit stronger as well.