It's just the difference between 'foul' and 'bad' as adjective. Foul' is a stronger intensifier than simply saying 'bad.' If we say something smells bad then it does not smell good, but if something smells foul then it smells truly awful. So just think of a foul mood as being a really bad mood.

The word "foul" is just a little stronger than "bad". Bad is rather a simple, general word, and "foul" is more descriptive.