I expect cavalry armour would get tougher and tougher, then design changes would move towards making it as light and flexible as possible once it becomes immune to everything on the battlefield. Infantry would also start getting more and plate armour due to more efficient production, but never as much as the cavalry since they have to march in the stuff. Something like the original Gendarmerie, a heavy lancer with a team of supporting mounted infantry and light cavalry, would form the core of most armies.
Hooked polearms might come back into vogue instead of pikes-if a cavalryman's horse is fully armoured, it can bulldoze through a hedge of points, but a halberd or bill can rip him off his horse and let him be killed on the ground.
Sieges are going to be a complete pain without gunpowder-any idiot can now have fortifications that are a right pain to assault. Castles aren't cheap, but they're easier on the wallet than star forts.
Swords will get pointier and pointier on the battlefield to strike at the joints on armour, likely culminating in tuck like weapons-alternatively, they might completely give up on anti-armour swords, with each soldier carrying a slashing sword for dealing with lightly armoured opponents and a bludgeon or rondel dagger for armour. Fencing could look very interesting if half-swording techniques remain important into the renaissance(assuming a renaissance happens-no gunpowder will change the mongol conquests, which might take out the black death, and Constantinople will take longer to fall without Turkish artillery!)
Butterflies from the 13th century onwards would mean the world would look radically different, even before we get to the increasing dominance of gunpowder in the 16th century.