More or less the same. There's scope for nuance. For example (2) sounds more like diplomatic talk, to me. I don't think people use (2) unless they're deliberately trying to stun the moment into sobriety; (2) has 'Shakespearean' overtones.
Whereas (1) is a goodly deal more pedestrian than (2).
A: "Well I think he's a Liar!"
B: "I know that he is an honest man." - I don't think that (2) would work here.
I think that 2. can be used less formally than Beryl is suggesting. Look at this example from the BNC:
Maybe he seeks a more thrustful mien, so that when he goes to his nasty little hutch in the City and glares at his neurotically blinking little screen and barks into his cellular telephone for another tranche of lead futures or whatever, he comes over as just a trifle more macho than we all know him to be.
While I agree that they mean more or less the same thing, I pick up yet another possible nuance or difference having nothing to do formality and do not insist that the nuance is anything other than my own:
"I know he is an honest man." He is a personal acquaintance and I have in my own life only experienced him in all our encounters as an honest man."
"I know him to be an honest man." By reputation in our community, people have certainly always said he is an honest man.
While #1 is an out-and-out declaration that I KNOW he is an honest man, #2 suggests that there is a possibility I may stand corrected. "I know him to be an honest man - from what I have felt, heard, been told". #2 is like saying, "As far as I can tell, he is an honest man".