Armor in D&D 5e mitigates damage by decreasing the chance you will be hit by an attack, not by affecting the damage of successful attacks (with the exception of feats like Heavy Armor Master). Resistance or vulnerability to damage is more a result of the anatomy of the creature being hit, not of the armor it is using.
So, it stands to reason that in order to make weapons more effective against certain armor types, it would be more appropriate to increase or decrease their chance to hit against different armor types. You can do this in one of two ways.
1. The Advantage/Disadvantage Mechanic
5e eschews flat bonuses based on circumstance, instead keeping things simple with this mechanic. For example, A dagger will have advantage on an attack against chainmail, since it can pierce through the rings (or so the logic goes). It's also easily implemented by assigning it based on damage type, rather than having a chart will different bonuses for each weapon. Slash attacks against chain have disadvantage, piercing attacks have advantage. Bludgeoning attacks against plate have advantage, etc. Now, you may come across situations where this can work against you. For example, rogues would be able to sneak attack targets based on what armor their target is wearing, even fighting one on one with a creature, due to weapon and armor choice. Still, I would recommend this mechanic over the alternative below.
2. Giving "To Hit" Bonuses
With this system, you actively give bonuses (or penalties) on the attack roll based on what weapon you are using. This actually isn't a new idea, and was included as an optional rule way back in the days of AD&D, as seen in Matt Colville's "The History of D&D, One Fighter at a Time" Series here. The problem with this system is that it's difficult to determine what bonuses to give to each weapon. You have to make decisions to quantify how much better each weapon is against each armor type, and you end up with a big matrix that you have to bring out every game, and it bogs things down as players adjust. And if they switch weapons, they have to look up the bonuses again, and all these look-ups bog down the game. This kind of system actively works againt the 5e philosophy by complicating rolls with a bunch of esoteric numbers and tiny little bonuses.
If you really want simulate weapon effectiveness against armor, I highly suggest you take the (dis)advantage mechanic to stick with the 5e principles and to save yourself the work of making a chart of flat bonuses that may turn out to be more annoying than fun to use.