Piston assy means you got a new piston and ring. I assume the saw had poor compression but the jug wasn't scored very badly, so they just polished up the jug and put a new piston in it. And yes, this will give you several more years of life from the saw as long as the rest of it holds together.
The 'worm gear' could be a couple different things. The first that comes to mind is the plastic gear on the crankshaft that drives the oiler pump. If they get hot or seized up, they can strip out and then you don't get any action from your oiler. A new worm gear will mean that your oil pump is spinning properly.