If you want to sound like a prissy English teacher use 'he.' If you want to sound like a normal person use 'him.'

Actually, they are both equally correct; however, I would argue that, as the nature of language is very fluid, whichever one is used more often is actually more correct. So you should use "I'm taller than him", because no one uses the other option. Good luck!

Actually, only "I'm taller than he (is)" is the only correct one. HOWEVER, most people use the other one - and they're wrong. So, you have a choice - to be grammatically correct or to use it like most people do. I prefer the correct way. It is my opinion that as long as you know the correct way, and can intentionally use it the wrong way (colloquially or otherwise), you're golden! Enjoy!!