From "Devil in the Dark", the Type 2 handle certainly adds "penetrating power", making the weapon useful against the silicon beast whereas Type 1 was somehow inherently incapable of hurting it.
TNG backstage doubletalk gives the bigger guns more "options" (in addition to power settings) and suggests that Type 1 can "barely" vaporize targets at its highest setting (eight lights glowing) whereas Type 2 and Type 3 have eight additional power settings beyond this basic vaporize. It's a dramatic conceit first and foremost: how much more firepower can you add beyond being able to completely make your enemy disappear?
If you can remove the rock behind which he hides, fine. But can you also remove the building in which his HQ lies? The hill behind which he is camping? Treknologically, you should be able to do that with your sidearm. But what does the ability do to drama?