Sounds very promising. 
Now to the nitpicking.
Armour:
As has been mentioned by others, plate armour doesn't protect very well against "blunt" weapons. Especially since blunt weapons like the war hammer and maces were developed especially against plate armour. From a balance perspective, the best armour, plate, doesn't especially need a bonus, while the weakest doesn't have one.
IMO changing, plate to have no bonus would be a good thing, since it's strong enough without the need of having a bonus. If it needs a bonus, it would be against cutting, since it's almost impossible to cut through a sheet of solid metal.
Since that is the same advantage mail armour has, I'd suggest expanding mails defense to cutting AND pierce, since, as long as the pierce is not in an almost 90° angle to mail or from slow velocity weapons, piercing mail is very hard to do.
Overall this would mean, that leather is the cheap armour, that doesn't need metal resources. Mail is the goto armour against most weapons (majority in your list is cutting and pierce) while plate stays the best overall.
Weapons:
- Axe and battle axe are classified as a cutting weapons, but they should be classified as a blunt weapon. You don't cut with an axe, you slash through armour and anything beneath it with its curved head. You could even slash through plate armour with enough force, which isn't possible with normal cutting swords. Especially late era battle axe was made to be used against plate armour.
- Bows, all bows are very similar, just getting stronger. Especially short bows should be much faster weapons then longbows. Also, that bows don't get strength benefits is very counterintuitive. Contrary to popular believe, bow warfare doesn't need much dexterity, since arrows were shot in volleys without aiming at individual targets. Strength on the other hand is very important for using the bow, since depending on the bow, sufficient strength is necessary to draw the string back. Thus the strength of the wielder of a bow directly influences an arrows velocity, possible flight distance and power on impact.
- Maul. Fun thing about the maul is, that it sometimes was an archer weapon. It was used to build fortifications against cavalry, ie. sharp sticks hammered in the ground which protected archers against cavalry charges and thus a readily available secondary weapon for archers to melee with. It's ok in a fantasy game though. 
- Spear. A spear is not a 2 handed weapon. Actually, a spear is almost always used with a shield. Ie. the popular shield wall. Spears are shorter and lighter then pikes, which ARE exclusively 2 handed weapons. But a spear, almost never.
As for piercing, daggers and war hammers should be classified as piercing too. Daggers, with their small crosssection can pass through mail and often through plate (where the plates come together, ie. at the joints, not through the metal itself ofc.). War hammers were designed to be plate armour piercing, which imo is the definition of armour piercing. 
On the other hand, spears and pikes... don't think they really can pierce through armour very well, since the weapons head usually was way too large though have enough force in a small area to breach armour.
Since slashing weapons would then be countered by 2 armour types, you could increase their values slightly.
Conclusion:
With that, broadly speaking, slashing would be useful against leather targets, piercing and blunt against plate, blunt against mail. Sounds like an ok balance to me then.
Sadly one important aspect can't be simulated with this weapon system, that certain weapons have intrinsic bonus against other weapon types. For example pikes are really really good against almost all infantery and cavalry since almost no other weapon then another, even longer, pike can reach them. But they just die to missile weapons, since they are unable to carry shields. Also if a pike unit gets outflanked, it's soooo dead. 4m long pikes in a tight formation has a turn rate that is really abyssimal.
But it's allready a much better system, so I can live with that.
Oh, and a small thing. Please call chain, mail instead. The name of that armour type is mail, the chain in chain mail only came much later. ^^
TL,DR:
Leather, good against nothing
Mail, good against, piercing and cutting
Plate, good against cutting
Axe, Battle Axe, blunt weapon
Bows, get strenght bonus
Spear, 1-handed
Dagger and War hammer, get armour piercing
Spear and Pike, lose armour piercing
