Evra's final ball was fairly poor (although he still got a goal/assist twice as often as Shaw), but that's hardly the only aspect of being an attacking fullback. He ran that left wing by himself quite regularly, getting forward far more than Shaw has ever done. His natural starting position when we had the ball was quite a bit higher, and once he had the ball he regularly took players on and created attacking positions out of nothing. His first instinct was always to drive us forward. Honestly, it was really only his lack of a final delivery that stopped him reaching an influence on the team like Marcelo or Dani Alves. All three could completely dominate their side of the field and cause chaos to the opposition, but the other two had the final ball to really make the most of it whereas Evra unfortunately didn't. He still played a huge part in keeping us on the front foot though with the way he created space and stretched the play.
Shaw sits deeper and will generally just pass it on as soon as a defender closes in on him. He helps out quite a bit with the build-up from the back and giving the midfielders an extra person to combine with, but the majority of the time he doesn't provide all that much actually going forward. He has the occasional period here and there where it seems to click and he's suddenly being a big influence, but 80% of the time he doesn't do enough.