As you'd probably imagine, 'a little bit' is less than 'a bit'. 'A bit of money' is a small sum, and 'a little bit of money' is even less.

Alyson is correct, they mean the same thing.

Your dictionary is correct; both phrases have the same meaning. Saying "I'm a bit hungry" means the same as "I'm a little bit hungry." Americans probably say "little bit" more frequently, while saying "a bit" is more common among British people.