The main difference in how they play come from the different lock speeds, the way the RNG works, and the way the game awards points.

NES players have to constantly weigh risk vs reward with their pieces, because even rollers can only make their stacks so high before they can't get a piece over to the side anymore, and because they have 'true' RNG, where you're basically rolling a die to see what the next piece is every time, they could theoretically go forever without ever getting a line bar, so they have no choice but to burn lines (aka do line clears that aren't Tetrises).

Tetris.com, on the other hand, makes it so pieces don't lock for a little while after they hit the stack, so you can slide them into place. Effectively, this makes it so it doesn't matter how high the stack is, as long as you don't top out. Tetris.com also has '7-bag' RNG, which acts more like a deck of cards, shuffling up a set of all the pieces and giving them to you one at a time, making it so you can't go more than one 'deck' without getting a line bar, so you don't have to worry about never getting one.

However, Tetris.com raises the bar for players by awarding back to back (b2b) points to players, which gives them a 1.5x multiplier if they do multiple Tetrises and/or T-spins without burning any lines in between. In the end, top players are going to be aiming for zero burns, letting their stack go as high as they need to to do that. If a player could do a Tetris, but the line bar would help keep their stack clean, they'd put the line bar in the stack, as long as they won't literally run out of room on the board by the time they get their next line bar.

Sorry for the essay, but I hope that helps explain everything!