dtgreene: I don't remember whether I mentioned it, but there are some games where the damage formula is different for different weapon types. You see this in some SaGa games (most notably SaGa 1 and 2, as well as SaGa Frontier), and in Final Fantasy 5 (but not other games in that series); this diversity of damage formulas means that different weapon types might work better for different characters or situations. (For example, in FF5 axes pierce defense and have a lot of damage variation, making them great on high defense enemies but not so good against weak cannon fodder (because you might get a low damage roll or miss entirely), while in SaGa Frontier gun attacks use the WIL stat, which is also used by magic, making guns the best physical weapon type for magic users.)
scientiae: As long as the correlation is not too abstruse I see no problem with different damage depending on the situation, opponent and protagonist. So a large mallet might have difficulty actually hitting a mouse (missing most of the time due to the rodent's superior dexterity) yet still squash one should the gamer luck out and connect.
One of the best changes the DS remake of SaGa 2 made, at least for someone not as familiar with the game mechanics as I am, is that the game actually gives you part of the damage formula for weapons, including the stat and multiplier. Hence, it becomes clear that Rapiers do their damage based off AGI rather than STR. Of course, it still takes a bit of thought or experimentation to come to the conclusion that equipping a robot with a whole bunch of them would be a good idea.
On the other hand, the ChainSaw's exact mechanics aren't so obvious, though in SaGa 2 having higher strength will allow you to saw apart more powerful enemies. (In SaGa 1, it's the reverse.)
Final Fantasy 5 doesn't have as much stat micro management, as any character with the same level, job, and secondary ability will have almost the same stats; on the other hand, the variation of damage formulas still allows advanced players to come up with interesting set-ups to maximize the effectiveness of less conventional weapons and spells. (Don't forget that FF5 is the FF game that introduced Blue Magic, which has many spells with unusual mechanics, like using the caster's HP in the formula, or Goblin Punch acting like a physical attack in some ways, but still having unique properties.)
scientiae: Another example is
UFO: Aftermath, where there are three types of ammunition, hard, soft and other, which are optimally applied against armoured and bloated enemies. So, for instance, FMJ rifle rounds will certainly pierce the flesh of a caterpillar-like psionic opponent but softer ammo will devastate it better, whilst softer ammo will tend to careen off hardened plates. The game also deploys different categories of protection (against inertia, energy, psionics, etc.).
edit: categories of defence
This works well if the elemental resistance and weaknesses are logical, and there isn't some easy way to work around this.
Phantasy Star 3, for example, does this wrong; enemy resistances don't have an obvious pattern, and in most cases physical attacks, which aren't affected by this, don't have that issue. (With that said, there is an interesting dynamic where most physical attacks are affected by enemy defense, but Wren's attacks are not, so it at least has something there. Plus, there is a technique that can cause instant death and is actually useful and cheap.)
There's also situations where you need to choose a specific type of damage to focus on, or a specific type of damage to take with you into a dungeon, and without extensive knowledge of the game (or very good hints in at least the latter case) is not a meaningful decision for the player to make. (I've encountered this in Lennus 2, where early game your main character can only have one element available at a time, and the game has elemental symmetry unlike the first game in the series.)\