From my recollection, interceptors tend to be armed with point defence weapons like vulcans, pd and burst lasers. Where as fighters are more often armed with proper sized cannons like motors, ion cannons, machine-guns (which double as PD) and IR Pulse Lasers. Other then that they are functionally the same, interceptors are often in larger squads and have less HP too, but having more fighters effectively make up for that in my opinion. Bombers are purely for attacking purposes and have missile weapons that are only usable on proper star-ships.
Interceptors are like having mobile point defence, having talons or wasps near a friendly ship can easily defend it form enemy missiles. Fighters are more like having an extra frigate and can easily take down larger ships in numbers. (Note, interceptors in large numbers can also do this, especially if they have a weapon that deals hight explosive or energy damage.) Bombers are mobile missile/bomb launchers and are honestly better then any missile weapon in the game because they never run out of ammo and take it at long ranges to the enemy.
The terms used for interceptor and fighter are not great in my opinion. Better to think of interceptors as fighters, and fighter as gunships/corvettes.