I'd like to point out that not all crossbows are that powerful. A two-handed arbalest definitely could, but a small assassin's crossbow made to just silently pierce a vein and deliver a poison could not. A one handed crossbow likely could, depending on the type of armor, bolt shot, and range. There's actually a well-respected Medieval myth that a crossbow bolt could shoot through a knight's steel plate, chain mail, go through him, out the chai mail again, back through the steel plate, and all the way through his horse.
Longbows were very similar in strength, but the arrows tended to be longer, causing them to lose more force as they were used for long-range bombardment, but were more accurate because the shape flew further (however this hardly mattered because they grouped together for volleys due to the fact that it still wasn't very accurate). Longbows, however, require years of skeletal structure-altering training and practice, as well as being harder to use close-up.
So yes, I'd say the crossbow piercing the armor is feasible. (And yes, I am an archery enthusiast, although I don't know everything.)
Note: Crossbows are ridiculously hard to reload. You can't even cock them by hand! You need a rope or other device, even for the smallest ones! Even then it takes forever
About armor: full steel plate with chain mail underneath it actually doesn't weigh too much, or hinder a lot. It only becomes a problem when the wearies and tires of battle get to you, but that will happen regardless.
Well, yes. Typically, I'm thinking of the Genovese crossbow-type, though, since if one compares the Longbow to the crossbow you really need to use the experts in their fields.
From everything I've read, Longbows were nowhere near as powerful as most crossbows.
However, they were faster to fire continuously, still did a damn good job(they could pierce chainmail like on the limbs and such, just not breastplate and helm and the like), and didn't need as much machinery for maintenance and the like.
Also, the difficulty of reloading the crossbow depends on the crossbow in question, the strength and skill of the individual, and the source you go to. One can fire a bow very very quickly, but it's exhausting, which was why longbowmen would typically fire once every 10 seconds. Crossbowmen could be at perhaps 30 seconds reloading time, if they were skilled and the crossbow wasn't more like a ballista.
@Andres: Okay, stop. Take a look at Conquistador armor, that time period and the like.
That is armor made to deflect bullets. It's not just about being thick, and also why the fuck would you use Warhammer 40k as a legitimate source for logic or armor capabilities? Leaving aside the bit where you ain't got the armor helping lift itself like they do, or the part where the shit it's made of is pretty damn high durability even for 40k, and then ask yourself why you think any of that would apply to this?
I could make fully encompassing armor that could deflect (some/most) bullets. What's easier, though, is "projectile-be-gone" with an enchantment that just causes it to veer away. But that's because I'm the god of Protection(don't get me wrong, a mortal could do it too, it's just not gonna be a given, if it's not magical, or anywhere near likely to work, particularly since they don't have guns to test it with). And Yaos wasn't.