Honestly, I vastly prefer homogeneous action resolution mechanics. To me, it makes for a much cleaner game and smoother play. And, that's not where you're ever going to find support for character individuality. Action resolution diversity gets you various mini-games from which the player can choose to use at the gaming table.
I think what you're saying is this:
Bob the fighter uses a short sword and specialized in it (applicable even in 2e, possibly 1e)
Rob the fighter uses a quarterstaff and specialized in it.
Bob is clearly better at using short swords that Rob, though both weapons do the same base damage of 1d6 (true in just about every ruleset of D&D that I know of).
They are differentiated by having a rule that lets the PC pick something to be better at.
The rule mechanics and resolution are the same for both PCs.
If both PCs got lost their weapons, they could fashion quarterstaves, and Rob would have an advantage that fit his character and the situation. This adds some flavor and pseudo-realism.
In too simple of a ruleset (no specialization, armor vs. weapon rules, feats or other choices), then both fighters are functionally equivalent and interchangeable.