“You are obsessed with using and maintaining a wide variety of conventional firearms. With each rank of the Gun Nut perk, you gain an additional 5 points to the Small Guns and Repairskills.”— In-game description
“The Gun Nut perk grants acess to more effective gun mods.”— VDSG Catalogue No.9270.
Having this perk will grant a dialogue option to the Lone Wanderer when starting the Reilly's Rangers quest by talking to Reilly, in which they will mention that they've heard that the Rangers carry "some serious firepower."
The perk image shows a missile launcher among the collection of weapons. The missile launcher is actually a Big Gun, and so it will not get a bonus from this perk but instead would benefit from the Size Matters perk.
The perk is needed to craft mods from scratch, not for applying or removing mods. One can remove a mod and apply it freely on the same kind of weapon, even if one could not build a new copy of that mod.
With the Automatron add-on, one point in Gun Nut is required to access the full range of craftable robot components at a robot workbench.