I think there's a misconception as to what this is, when players feel 'weightless'. If they were weightless, you'd be able to change direction instantly, and there would be none of that sense of pushing left but the player still going right. The responsiveness would be instant when doing a 180 turn.
There's far more inertia and momentum in FIFA than PES. In terms of getting a player up to speed from a standing start, there's no question that it's far more apparent in FIFA that a player has inertia.
The problem is more one of responsiveness - of how players actually accelerate or decelerate. At times when you move a player in FIFA, you forget that the players move using their legs. There are definitely certain animations which are mo-capped and therefore are convincing in the way a player can take a sidestep and then continue forward. But, during more generic player movement, there can still be a sense that it's the body dragging the legs around the pitch, rather than the legs pushing the body around. Like a chinese martial arts film where the character is on wires, running along the rooftops.
I would definitely say, though, that it's a lot better than it used to be. I definitely feel like the dribbling is so much more convincing than it was a couple of years ago - I've done some fantastically convincing little shimmies, or deliberately avoided a tackle, using very deliberate footplanting.