I think tumpliner may have misread your question. There is only a subtle difference between these two sentences and neither one is grammatically incorrect. "That's what he's got" emphasizes his CURRENT possession of whatever you're talking about. For example, if my child is in a toy store with me and sees another kid walk by with a toy, she might say "I want that one because that's what he's got!" At the same time, the person who possesses the object/thing doesn't need to have it with him or her. For example, I can also say "that's what he's got" if I'm telling my friend about another friend's new television. The television is at his house, even when he's gone and I can still say "that's what he's got." "That's what he got" does not emphasize currently--it only means he got it SOMETIME IN THE PAST. He might not have it any more.

No-- the second one is poor grammar. It helps to write out the sentences without apostraphes to see the difference: That is what he has got. That is what he got. So the difference is the second sentence has dropped out the has-- which is necessary to be good English grammar.

I remember my teachers in school telling us "got" isn't a word. In oral English we use it all the time though. Just don't write it down anywhere a teacher might see it. The two sentences are really close in meaning. Let's try some examples to find any differences. Does he have a truck? "That's what he's got." Did he buy a truck? "That's what he got." I need a spoon not a fork! "Too bad, a fork is what he's got." I wanted pizza not Chinese food! "Too bad, that's what he got." "That's what he's got" is the same as "That is what he has got" the "has + past participle" makes this present perfect tense. "what he got" is simple past tense.

difference is good grammar vs. bad