TL;DR: big tech level differences and anachronisms are not stable. Your "barbarians" will either end up with medieval-style weapons and tactics and a suitable society needed to support them, or they'll be crushed, or they'll be an irrelevance.
So, obviously, military technology improves given a few hundred years. Better weapons, better armour, better horses with stirrups and horse shoes and their own armour... all these things give the more developed army a big advantage over the less developed one. There are perhaps other, less obvious things, like improvement in medical care (not much in this case, I suspect) and improvement in farming and nutrition, which will help produce tougher, healthier soldiers. Better construction techniques give better fortifications. Better engineering gives better siege weapons. Better boats lead to better logistics.
Your "barbarians" are clearly at a huge disadvantage if they try to operate on foreign soil... maybe they have some home advantage, but if they have farms and settlements they're very vulnerable. They know this. The medieval society (or societies) know this.
Those war hammers, flangeds and spiked maces, and various things like bec de corbin that are designed for fighting armoured oppoents? They obviously won't have them, or even consider them to be useful before they've fought the sort of people they were designed to be used against. Assuming they're not slaughtered promptly though, they'll start getting them. You've hopefully not created a planet of hats (and if you have, hang your head in shame), there isn't going to be "the medieval country" and "the barbarian tribe". There will be many of each, and some will fight, and some will trade. Your "barbarians" will learn about the design of weapons to pierce and crush armour, and how to get them. They'll learn about use and creation of armour. They'll get better horses, and improved tackle for them.
In due course, they'll be well equipped to fight the medieval-level opponents, because they'll have pretty similar kit all of their own. You'll miss the style of topless greatsword wielding conan-alikes in fluffy hotpants, but on the other hand you won't get to see your conans all shot to death with crossbows, ridden down by heavy cavalry or beaten and hacked to death by better armoured opponents. If you can't beat em, join em.
There's no secret weapon or magic technique that'll let them stay the same and come out victorious, unless your universe literally has magic and your "barbarians" aren't adverse to using it. You wanna deal with heavy cavalry? Then look at how medieval warfare evolved... Lansknechts, Swiss Pikemen English Longbowmen, Geonese Crossbowmen and the like. A cursory glance at a short list of infantry vs cavalry techniques in the middle ages might give you what look like some ideas, but the infantry in question are moderately well disciplined and equipped medieval soldiers that aren't really equivalent to the folk who fought against the Roman legions.
I have just remembered the "gallowglass", which apart from having a pretty bad-ass sounding name are often imagined as a big guy in a kilt with double-handed sword, like a sort of conan for colder and wetter climates. The original gallowglasses were even descended from vikings. Don't be fooled though; those guys were well equipped and had high-quality armour. Not simple steppe warriors!