In principle, there is not much difference in the amount of damage between a bullet thru the gut, an arrow thru the gut, or a sword thrust thru the gut. The piercings are comparable. Roughly 1d8 damage.
What makes the gun differ is the attack, not the damage. A bullet is more likely to hit.
The gun is a simple weapon that most people can use.
The bullet is more likely to pierce − thus ignore − armor.
The biggest change that firearms introduce into gameplay is, combat typically becomes ranged, and cover becomes more useful than armor. This distancing of combat reduces the visceral quality that makes melee combat more exciting and more entertaining. To introduce firearms dramatically changes the nature of the D&D game. Guns have their own kind of entertainment, such as suspense because a bullet can come from anywhere at any moment. But it disresembles melee combat.
Maybe the best way to handle firearm mechanics is, instead of rolling an attack d20, the target rolls a Dexterity (Reflex) save. The armor is less relevant. Mechanics can assume the shooter has general competence to aim the gun in the direction of the target. It is up to the target to become less easy to hit, by moving around and taking cover.
For modernesque heavy firearms that do extraordinary amounts of damage, these are more expensive. To put these heavy weapons in the magic item category or its technological marvel equivalent can gatekeep these well enough.
But in all cases, having firearms use the Dex save mechanic rather than the attack mechanic seems to work realistically enough and within gaming balance. In other words, firearms are more like a magic cantrip or magic spell or consumable magic item, even tho they are Martial technology.