Xbows have no bonus against the armour, they just have a high base attack. Personally I am a horse archer. I put the kergits to shame. When your proficiency hits about 350 and you have horse archer 5, master the firing while moving and picking off other riders becomes no problem. I use the bastard sword for dispatching the infantry. One thing the guy above forgot about the bow... Our power draw adds to the damage. Max 40% boost. The draw requirement of the bow is also added. a bow with draw 8 and 40 damage, is actually boosted by 120% which means the damage is 88. Higher than any crossbow. You just need to invest the skill points and the time to get the proficiency high enough to use the bow effectively. Higher the requirements, the more proficiency you need to be accurate. Crossbows are easier to learn, but the bow "eventually" is much stronger. (and faster) That said, you don't need the best bow. Get a good bow and upgrade as your proficiency increases. Putting an arrow in their head is the goal, not putting an arrow through the head if you get what I mean.
In one of my campaigns I have eliminated the kergits and own half of the Vaegir, Sarranid and swadian territory. My 5 heroes have 6 or higher trainer, making them weaker fighters, but I can upgrade over 100 troops with all of them leveling once a night to the second last tier. In times of peace I gather recruits, and stockpile then in my towns. Start of a war, I grab a full complement, and by the time I arrive at the enemy territory, I have professional troops. I use a ratio of 10:1 cav to archers. No infantry. Utterly destroys any AI army. ( I only have archers so when I run out of my 102 arrows in a siege (all head shots, or body on soldiers I know I can 1 hit kill) I can pick up the arrows from my archers that got picked off) I play on the max difficulty, (rated by the game as 149%) and with this army and strategy, I kill most AI at 10 to 1 casualties, and 25 to 1 deaths. My buddy would swear by the crossbow, but I would say that the bow is better, just more investment necessary.