Sorry, I must have missed this one. They're both fine, though:
1. "Are you any better today?"
You could use this if you were asking a sick friend if he felt any better.
2. "Are you any better than him?"
This has a couple of possible applications:
a) To express a doubt about someone's behaviour, in relation to someone else, e.g.
MrQ has posted an abusive message about MrP.
MrP has posted an equally abusive message in return.
Later, MrP is telling Blacknomi all about it:
"And I tell you, Blacknomi, I'm shocked. Shocked and appalled. To post such a message in a public forum! It's outrageous."
"But didn't you say you posted an equally rude message in return, MrP?"
"Indeed I did, Blacknomi, indeed I did."
"Well, you'll forgive me for saying so, MrP; but are you really any better than him, in that case?"
"Probably not, Blacknomi...But it was fun while it lasted." :)
b) To enquire about someone's ability relative to someone else. It has a slight air of doubt:
"I hear you like skateboarding, MrP."
"That's correct, Blacknomi."
"MrQ likes skateboarding too."
"Indeed he does, Blacknomi."
"I hear he's quite good."
"Oh, he's not that good, Blacknomi."
"And what about you, MrP? Are you any better than him?"
(The straightforward 'are you better than him' is more neutral.)