What makes a cat nuzzle aggressively?

Our neighbors' cat spends a lot of time at our house and when he arrives, he has to rub his nose and cheeks against our chins really hard. But often, later on, he stretches his body up toward us when we're standing and when we lean down, he grabs our shoulders/neck with his paws and pulls, digging his claws in. Then he can't seem to get close enough to our faces no matter how hard he's pressing, taking moments to nip our chins as if we aren't responding correctly. If we pick him up, he gets rougher. What's going on and how should we respond?