I don't think it would change much, compared to the middle ages. Let's compare!
Depending on which time period you refer to, my answer differs. For most of the middle ages, cloth armor like the "gambeson", a padded torso and arm armor was dominating for most soldiers. Steel helmets were worn whereever possible. If you had the money, chainmail was worn. A crusader for example might have worn a gambeson with chainmail on top. Later, in the 15th century and beyond, platemail became the dominant choice of armor. Contrary to common believes, platemail allows for great mobility and agility if forged right.
Cutting through a chainmail with a sword is veeeery hard to do (i would say almost impossible in real combat). Also, piercing with a spear or arrow becomes difficult, especially when there is padded armor underneath. While arrows with special tips had good chances, i think melee weapons had a tough time. Blunt weapons like warhammers or maces did better, and often a sword blow would crack or shatter the opponents arm without actually piercing the armor.
Yet, in "real" combat, blows would most often not align perfectly, the opponent would deflect or dodge most of the power, and so armor protected very well from the "pricks" and half-hits that were most common (and would still have fatally wounded an unarmored opponent).
With introduction of platemail, the game stepped up a bit. Platemail is pretty invulnerable to most melee weapons. Anything below a halbert wielded during a charge should not have been able to puncture a breastplate. Given that the wearer is not helpless on the ground of course. Special weapons (warhammers mostly) were invented that allowed to puncture platemail, but most weapons were completely inefficient against platemail. So combat techniques adapted. The warriors shifted their focus to the weaknesses of the armor. Knees, elbows, wrists, neck, armpit, groin, hands and face had to have weak points, so the wearer could move at all. Fighters focused on hitting these weak points. To make that easier, "sword wrestling" was in heavy use, aiming to throw the enemy to the ground or immobilize them, so hitting the vitals became possible.
Even when crossbows and musquets spread on the battlefields, platemail remained the armor of choice until the late 17th century.
Now, how would modern technology influence all this? Whew. On the weapon side, it's possible to manufacture more durable, lighter (or more likely: better balanced, since physical impact depends on weight) and more flexible weapons. I don't think the sword would vanish, it proved to be the weapon-of-choice for millenia. We could have sharper edges, but i don't think the weapons could improve that much more. look at modern firefighter axes and tools. They are sturdy as hell, and razorsharp, but an axe is an axe...
On the armor side, i am thinking carbon nanotube armor. See here: Carbon Nanotube on wikipedia That stuff can withstand almost any physical blow (bullets, even), and is super lightweight. It's far superior to most materials we can actually use, but let's assume we have that stuff. Because i think using our "more common" modern materials will not change the essentials of what i am saying, just making it more complicated.
So armor still follows the same principle: you cannot penetrate platemail at all, but you have to go for the weak points. But these days, the weak points can be layered with kevlar and other modern materials. Your visor can be made from transparent aluminium or armored glass, so that's even not a weak point anymore, AND you get perfect vision. Try wearing a medieval full helmet, it's horrible. You don't see sh... shiny things.
Sooo. I think battling other combatants will be primarily about getting them to the ground, and exposing their weaknesses for a good, full-force thrust to their weaknesses, killing them like that. Maybe nets, bolas and the like will see a revival? Also, armor-penetration weapons like the warhammer or the halbert will be veeeery common, while axes and swords might be less frequent.
Sadly, i think bows are gonna have a bad time.
lastly, don't underestimate that armor is expensive. So if, on the modern battlefield, not all combatants are armored in fullbody platemail, it looks rather different. In that case, the balance of weapons will shift a bit.
TL;DR: I think it will more or less resemble battlefields of the 16th century, minus the firearms. :)
Afterthought: do tasers work, if i charged them with muscle power?