I play with both my cats using a wand with various toys on the end of a string (never hands), but they do so very differently.
My Bengal likes to stalk the toy until it stops moving (usually just out of sight) and then pounces on it. If I pull it before he jumps, he stops and waits. If I pull it during his jump, he will chase it for a second at most.
My Savannah likes to chase the moving toy contiuously until he needs a break to catch his breath, then he will stare at it until it starts moving again so he can resume chasing.
In both cases, they usually stop when they catch the toy and walk away until I hide it or move it again, respectively, though occasionally both will carry it off to a corner to lick and chew on.
There is also a difference in how they play with balls, though I can't as easily describe it.
Is this difference just an individual personality thing, or does it have something to do with their different wild ancestors?