Weapon damage is a factor of your skill, perks, and any equipped items you have which boost weapon damage. There's no "leveling" going on. If you have two weapons that do the same damage now, and they are both the same kind (one-handed/two-handed/bow), then they will always do the same damage even if one is one your hand and the other is in your closet at home.
A special note: Some unique things ARE leveled. If you get a unique artifact at level 20, then it will be worse than the same artifact if you got it at level 50. Personally I consider this a bug because I cannot even imagine any developer thinking this was a good idea. A fix should be available to PC once we have the construction set, a continuation of an identical fix in Oblivion that solved the same problem.