Wanted to be clear, not saying don't do it, just realize you're going to slow down the game a bit until players get used to it. AD&D failed on this rule because weapon types were so complex they required a chart to cross-reference. In my brainstorming on weapon types, people talked about art and literature from medieval times where a warrior carried multiple weapons: e.g. dagger, short sword, big sword, and mace. I thought it'd be neat if we could make weapon choice more realistic.
And we can, but the further you go, the harder it is to work into the game without the #1 killer of combat excitement: delay.
For example, a dagger is a poor weapon against a warrior in plate with sword. Equally skilled, you're probably going to lose. But put that same dagger close up, in a grappling struggle, and the dagger is great for finding the weak spots where the armor ties and hinges together. Do you give the weapon advantage on damage when used in a grapple against plate, or any armor, and what about no armor? A flail is an unwieldy weapon, but against a shield, it is designed to wrap around and make crushing contact. Do you give it advantage against shield users only? Even if you do (and AD&D did, bonus vs. shields), will your players remember it, and will you remember it if your monsters use particular weapons? What about no armor at all? Does it make sense my weapon does more damage when you're wearing plate armor but less if you're wearing only your underwear? Do we give damage advantage to all weapons against those who have no natural armor at all? Does even 1 point of natural armor count, and what about monsters? Does the skin of a giant lizard act as leather armor, or do our rules only apply to manufactured armors and weapons? What about an animated statue with a mace weapon, does it count as a mace or because it's carved does it not?
You get the idea, one question leads to another and another. And the last thing you want to be doing is arguing with your players. So if you do implement it, and it's a laudable goal to reward players for making a conscious choice on weapons, keep it simple. Make a list of weapons that deserve bonuses and run this by your players and the gaming community. Don't include monsters. Decide how you want to handle unarmored situations (and what counts as unarmored).