Essentially they convey the same information. However, in the following kind of situation:
A: Nobody thinks that.
B. (Well,) I happen to think..
the second one might be used. It puts more emphasis on yourself as an exception and acts as a kind of negation of the previous statement.
I would just say "I do." I don't like to waste words. :)
You can do this with other verbs as well..
A: Nobody exercises these days.
B: I happen to exercise every day! (I do! I am an exception!)
It can sound kind of pompous as well if you inflect it differently.
Well, /I/ happen to..