Fallout Shelter: Weapon types, do they matter?

From this post, based on the organized list of weapons, it seems that, in general, Heavy weapons will do the most damage, followed by Energy weapons.

It doesn't seem like the weapons are classed by type due to statistics, so much as they are classed based on what they actually are. A Heavy weapon and a Pistol that do the same damage should be equally effective when used by a single Vault dweller.

As to your question about stats across gun types, critical chance is determined by a Vault Dweller's Perception, while raw damage is determined by their Strength. Shot speed is determined by Agility. Therefore, these stats rely on the stats of the Vault Dweller who is equipping it. The type of gun does not determine this. If anything, it would be specific to each gun.

It depends on the stat of the person, how powerfully the weapon is used. There won't be a difference except in damage.