“He is authentic”
This means genuine, rather than a copy or a fake. If a man wears cowboy boots because he spends all day riding horses and roping cattle, then he is an authentic cowboy. If another guy wears cowboy boots because they look stylish, but he works at a coffee stand, he is not an authentic cowboy. If you say someone is authentic, this usually relates to the whole person, in the sense of “what you see is what you get”. They are for real. They are not putting on a show. They appear on the outside as true to their person on the inside. They would act this way even if no one was watching.
“He is sincere”
This usually relates to a specific topic that the person is sincere about. He is sincere about reversing global warming. He is sincere about being a good parent. This is when someone has strong emotions for a topic, and those strong emotions are what they really feel, not just what is fashionable to claim that you feel. You would usually comment about someone being sincere about something. I don’t think I would ever say just “he is sincere” because it doesn’t show what he is sincere about, and I don’t think someone can be sincere about everything! You could say “he sincerely loves his wife” which means he does it because his heart is in it, not just because it’s what you’re supposed to do when you’re married.