I've seen sites that say crossbows were virtually useless against armor, to the exact opposite, crossbows being so effective they were outlawed by the Church and caused heavy armors to lose popularity.
So just how well would an armored individual fare in a battle that employed crossbowmen?
hi! Not discouraging anyone from providing more information on your question, but FYI there have been several posts asking about the effectiveness of armour against various weapons, including crossbows. I've rounded them up here, as a handy reference in case we decide to create a new section for the FAQ. They're in roughly chronological order; you can skip the first one, which predates crossbows:
Just how effective and were slings as weapons in ancient warfare and how were they used?
A question regarding the Byzantine Klibanion (κλιβάνιον)
How well did medieval armors protect?
The Battle of Agincourt - Breaking down a few inaccuracies
The bow is better than the musket - why did Napoleon not use archers?
Crossbows were very effective against opponents in armour and during castle attacks and defence. The drawback was that they were longer to reload than the British long bow for example.
It meant that the French for example (who employed a lot of Genoan crossbow mercenaries) would fire one volley of crossbow fire for every 2 or 3 English longbow volley. It basically made most battles of the one hundred year war between them a race against time for English longbow men. The English would emphasize their archers a lot and go a very long way to protect them as they were their most important asset. The French relied a lot on heavy cavalry (their nobility was numerous and rich and well armed) that would of course charge straight to the British archers as they were very vulnerable and were the main English asset. So basically if the English made enough damage to the French knights before they came into contact it was a English win, if the French knights successfully engaged the English archers, it was a French win. The goal was therefore generally for the English to litterally shower the French knights with arrows to hurt them or their horses before they came into contact. For that the higher speed and the good range of the longbow was ideal in open field.
The English tended to have a harder time to take castle though as crossbows favored fewer but more precise volleys and with a much higher piercing power, which is better if you happen to have all the time to reload (as in if you are sitting in a castle behing huge walls) and are shooting as enemies that are trying their best to protect themselves with wooden planks, shields and such at long range.
So to answer your question crossbows were definitly better if you only take into account the piercing power. But bows were more a "quantity is a quality in itself" type of weapons and it worked very well too.
The British were also able to field a huge quantity of archers (they would sometimes make up half or more of the English army) because every British peasant was required to own and be proficient with the long bow throughout his life, meaning the Brits always had a force of longbow men ready to be drafted according to the needs. The crossbows were more expensive and more complicated to use, meaning you would trust mainly trained mercenaries with them (the Genoans were reknowned specialists) so you could never field as much crossbows as bowmen.