Well, of course it was the 16th century (or late 15th) arquebus that was replacing the crossbow, not the earlier handgonne, and I'd assume there was a good reason for that. I don't know if it was the rate of fire or the penetrating power of the gonne that was unsatisfactory; the barrels on them seem awfully short, and barrel length is a major factor in a firearm's efficiency.
For the crossbow, the rate of fire varied massively--the rule of thumb being that the more powerful the weapon, the more time it took to reload. The zhuge nu, although it wasn't used in Europe, could discharge its ten-bolt magazine in under fifteen seconds, and would have had a sustained rate of fire of between twenty and thirty bolts a minute. Of course, a zhuge nu was about as weak as any medieval ranged weapon could get; against even lightly armoured targets it didn't have any real killing power (bar a lucky shot through the eye) beyond maybe twenty metres. I'm not sure it would have worried a man in full Milanese even at point blank rnage. Suposedly, the Chinese took to smearing its bolts with poison, because otherwise they often wouldn't penetrate far enough in to make a kill. Besides, it was wildly inaccurate since it was fired from the hip by pumping a lever back and forth, which would probably have imparted a rocking motion to the crossbow.
At the other end of the spectrum, the heavier, metal-bowed crossbows used during sieges to "snipe" at the enemy from very long range were extremely powerful, the windlass mechanism allowing them to make use of more than a tonne of effective draw power. This made them extremely dangerous and quite accurate; at close range they probably could reliably pierce plate. But it would have taken ages of toil (more than a minute, IIRC) to draw the string back, and the weapon itself was quite heavy, so that they usually would be reserved for siege warfare (like the Greek gastraphetes, although it was nowhere near as powerful).