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One system that I think did this really well is BECMI D&D - it has five levels of weapon proficiency, not including non-proficiency: Trained, Skilled, Expert, Master and Grand Master. Each level of proficiency does more than just static bonuses - it adds new abilities to the weapon.
The Normal Sword, for instance, gives a bonus to AC vs an increasing number of attacks (Skilled gives you a bonus to AC vs one attack; Grand Master gives vs three), the ability to make a saving throw to deflect an attack that would otherwise hit a certain number of times per round, and increased damage per hit. The Battle Axe, on the other hand, gives bonuses to AC and increased damage at a different rate than the Normal Sword, but instead of allowing you to deflect attacks, it allows you to daze your opponents unless they succeed on a save.
Naturally this stuff is probably more powerful than we really want proficiency to be, but it does give some ideas for making different weapons feel genuinely different and for rewarding higher levels of skill.
It would likewise be nice if something like this could be added to skills - currently, going from Expert to Master just feels like a +1 bonus, which is honestly just boring.