Water transfers heat better/faster than ice. Also, this may seem counterintuitive, but it when water freezes, it actually gives off some energy, warming up things around it. Not enough to melt the ice back into water, but just a bit. Salt water also has a much lower freezing point than "fresh" water.

So if you salt the ice on your driveway, you're lowering the freezing point so that the salt + ice will melt back into saltwater and flow away from wherever you're putting the salt.

If you put salt on ice for a dessert, you're letting the water stay liquid at lower temperatures while the remaining ice brings the temperature down. This draws more heat from the dessert, keeping/making it cold.