lichess.com does have a snappier feel. Using the chrome performance tools it seems that lichess takes about 2ms to go from mouse click to the piece selection repaint and lichess takes 12ms. (screenshots of traces: https://imgur.com/a/iUkGm8F)
Lichess also uses zero-time premoves and chess.com uses .1 second premoves. I think these also contribute to lichess feeling snappier.