Long Range: Hunnic composite bow (toxton), sling vs. Bandolet Wheelock carbine, Luk
Before I give an edge let me talk about how each weapon and how it is going to be used, since some questions have been asked in regards to a couple of them.
Lets start with the sling. In my Cataphract Bio I may have accidentely implied that the sling was used on horseback. This is false. I have yet to find a single instance of a Byzantine slinging from horseback. It was just given as a backup to archers. In the scenario I see two ways that it can come into play. The first opportunity for the sling to be used is at the beginning of the fight when the two teams are about a couple of hundred meters away from each other. Here they could dismount to hurl lead at the hussars, who would have to retreat out of the slings range or rush into the battle . Presumably once the hussars got in a certain range the Byzantine would get back on their horses and switch to bows. The second circumstance is if an archer gets dismounted in battle and has either run out of arrows/broken bow. Now he has a backup weapon to switch to. I am going to split up the damage review into two parts: man and horse. For the man I really can't see it doing much. The armor of the Hussar is too thick in too many places for the lead bullets to do much. Sure he may suffer some bruising and at worst, a fractured bone, but I cannot see the sling doing much damage in the places where the hussar is armored. Of course if the bullet got a lucky (or extremely accurate) shot to the uncovered portions of the neck , face and eyes then yes that Hussar is going down, but that is pretty much a given. The horse on the other hand.....
Hussars did not wear much in the way of horse armor, preferring speed over protection. As a result a lead bullet to basically any part of the horse is going to be completely devastating. A horse isn't going to be able to run at all with a broken leg or a fractured chest which would take the mount away from the hussars, putting him at a disadvantage. This is assuming he survives of course cause if you break a horse's leg while he's in full gallop then his rider is going to be sent flying forward when the horse collapses, possibly breaking his neck. In conclusion the sling won't be the greatest weapon in this fight but it certainly has potential to help the Byzantine greatly.
Now onto the Hunnic Composite Bow: As I noted in the Byzantine bio, Byzantine Cataphracts were expert archers and will be bringing a high degree of skill to this match-up. They worked in combo with the lancers to ensure maximum devastation to the enemy formation(remember the point of heavy calvary were to break up these lines). I can see their armor piercing arrows making it through the weaker points of Hussar armor but their real advantage here will be shooting out the Hussar's horse. The best of the Byzantine's ranged weapons.
The Tartar bow is the Polish Hussar's first and most common of the long range weapons. It has roughly the same range as the Hunnic bow,has a high draw, can be used from horseback and is quite accurate in the hands of an expert. Unfortunately the Hussars were noted for their deficiency in the field of archery, and while good Polish archers did exist in that time frame they are the exception rather then the rule. Further adding on the Hussars problems in this category is their lack penetrative arrows, which means that only a divinely guided shot is going to kill anything. The Byzantine's horse is protected by steel or ox-hide Lamellar, which was quite resistant to arrows. Thus this weapon may actually be the weakest in the fight.
The Wheelock carbine is where the Hussar shines in this match up. While it has many problems such as; it is prone to misfiring (especially in bad weather), takes a long time to reload , has a small range when compared to the weapons in the rest of this category and is practically a one shot weapon, it is the only ranged weapon here that can tackle the big issue: armor. While I am not completely sold on it penetrating the many layers that make up the Byzantine's chest armor , it can penetrate just about anywhere else including the Byzantine's horse's armor . It can be accurately used from horseback and pulled from the saddle fairly quickly. The bandolet is the best weapon of this range.
Overall it is really hard to give an edge in this category, as the Luk(bow) is going to have relatively little effect on the outcome of this matchup while the Bandolet will have a high effect. The Sling's contribution will be slightly greater then the Luk's while the Hunnic bow will be in the high-middle range For all of its faults, I think that the Bandolet is powerful enough to give the Hussars a very slight edge in this category.
Edge Polish Hussars
Mid Range: Marzobarboulon , Kontarion lance vs. Flintlock pistols, Koipa lance
It is a bit humorous that the main weapons of both warrior, the two lances , are both going to be near useless in this match up due to the excessive amount of armor. The best either is going to be able to do is knock the other off their horse, which is pretty unlikely given the superior saddle of the Hussar and the brittleness of the Kopia. Both have small advantages though; the Kopia has nearly three times the length of its rival, and the Kontarian can be thrown. A thrown Kontarian will likely do the same amount of damage that a hand-held one will do, i.e. none at all, but it will still be an effective distracting tool. I can't really call this one.
Pistol vs. Marzobarboulon seems like a bit of a miss match but the darts do have some advantages . For one they don't jam and if held by a professorial are surprisingly accurate and its possible these could be thrown into the eye, chin or neck. They also have a bit of a surprise! factor, and can be hidden behind objects such as a shield or even in the sword hand and thrown. Even if they don't bore through the Hussar's armor I guarantee he'll still be feeling the sting, and the Byzantine will be able to take out his horse with it. Same could be said about the flintlock though, and unlike the darts these will be able to get through the weaker parts of the Kataphract armor (like the face). For more lethality I am giving it a edge.
Hussars are in the lead here, although not by much
Every single weapon here has problems with armor, and so this category will not be decisive one. The Spathion and Sazabla are both slashing weapons, and in this battle the only way to get kills by slashing are to sneak up and grab your foes throat, ripping off the helmet, and slitting it. Not going to happen very often(although if timed well it may be able to hit the weak points of someone's armor). The Paramerion and Pallash are better at thrusting, but neither is going to really have the power to get through even riveted chainmail. Finally the Koncerz, though unwieldy , tips the edge in Hussar favor. Due to its length the Hussar's could and would use this as a small spear, and being hit with it would be painful to endure.
again slight edge Hussar for having more options
Extremely close Range: Byzantine Dagger vs. Polish Hunting Dagger
Obviously in a mounted battle these two weapons will hardly ever see use, and when they are their effectiveness will be limited by length and lack of armor penetration (only way either are getting kills is through precision strikes to the weak points of each's armor). The Polish hunting knife is a better knife then the Byzantine dagger, but its rarity hurts it here. The Byzantine dagger would have been available to every man, and for that reason it gets the edge
Special: tzikourion(axe), Bardoukion and other maces vs Czekan axe , Horseman's pick
Both warrior's arms are nearly completely covered, which is important since this section may be targeted first in a duel, as it holds the weapon. I am giving an edge to the Hussar for having a single Gauntlet, although
as you can guess this edge is small.
Legs: leather boot, greaves(podopsella) , solid steel Lamellar skirt(kremasmata) vs. Tasset, leather
This edge is fairly simple. Edge to the Byzantine for more coverage.
Blocking: Kite Shield vs. Dueling experience
The edge is obvious, but limited. The Shield is too heavy and cumbersome to do much maneuvering in this battle, although it does give some extra protection to the Kataphracts left side. Dueling is going to help in sword to sword or hammer fighting, further limiting the Byzantine edge.
Horse Armor: Stirrups, Basic Saddle, Lamellar(Iron, Steel or Ox-hide),plate headpiece vs. Stirrups, Oriental style saddle, No horse armor
The saddle has advanced since the Byzantine's time and the Hussar saddle will keep him from getting knocked off the horse a bit more then the Byzantine's will. Although the lack of armor of the horse is going to enable them to be much faster then their slow counterparts, it makes the horse an excellent target.
Byzantine Tactics vs. Hussar tactics:
Yet another tough category to decide. Neither charge is going to be really effective due to the strength of the armor and weakness of their lances. The harrying tactics may be effective, but it leaves the archers vulnerable at the same time since the Hussars attack in waves. Both are going to conserve energy as much as they can, and both avoided heavy calvary in history when they could. In the end the mass of Byzantine military manuals (sometimes called the Byzantine art of War because of the way they methodically list strategies to almost every unit type/ nation that surrounded Byzantium at the time) and hammer blow tactic give him the edge, although it will only be a slight one.
Motivation/Morale: Made Peace with God before battle, System of rewards and punishments, warrior saints, Army and Emperor vs. Confessions before battle, Patriotism, Social Maneuvering, Pay
Both of these sets of warriors are highly motivated, and I cant really give an edge.
Rules Of Combat: Last Prayer, Last Meal, trot until within a close distance, charge only at last possible moment. vs. ranks, treated fellow Hussars as equals, conserve energy, forbidden from leaving battle formation
Another even X-factor Category. In my view neither of these really hurts or helps either them enough to give an advantage to either side.
Discipline: Byzantine vs. Hussar:
Both sides are neck in neck throughout this category. Both cultures had experience before entering the army, both were trained rather extensively, and both were surrounded by enemies. The Byzantines get this through two somewhat small factors: A. they were expected to train off campaign, and B. Byzantine Cataphracts were recruited based off wealth and ability, unlike just wealth for the Hussars. slight edge
Martial Arts: eclectic fighters, War Sports, Some evidence of classical Greek martial arts still in use vs. eclectic fighters, "War Sports" , Fencing, duels
Both had deadly war sports, whether it be an emperor-killing form of Polo or extreme jousting, and both were eclectic fighters. Fencing and dueling give the Hussars a much greater advantage then the Greek martial arts(which they may not have known about) give the Byzantines, as you can't really wrestle off horseback and in 106 pounds of armor. Full Edge Hussar