The term "musket" seems to have originally referred to a larger weapon than an arquebus, specifically designed to pierce good quality plate armour. I'm pretty certain the weapons became distinguished in the later 15th century, as firearms started to become more common, and armour technology responded.

That said, early terms for firearms are usually hard to pin down. The weapon categories blur into one another, and authors don't use the words in the same way. It may be misleading to see "musket" and "arquebus" as distinct weapons, and more of a case of one word taking over from another. Certainly later "musket" were more the size of earlier "arquebus" than early "muskets". I note that the book I'm currently reading (War in the Middle Ages) includes various terms for 15th & 16th century handguns, but doesn't use "musket" at all.

Handgun technology apparently didn't change all that much between the 16th and 18th centuries (someone else may dispute this). As DonaldFDraper says, powder could make a big difference, and it was developments in powder that really made the difference in the 15th century. Manufacturers got closer to the ideal ratios for the components, making more powerful powder. The process of corning gunpowder made the components less likely to separate, made it more resistant to damp, and made it burn better and more slowly. The first two features made it more practical for handgunners while the last increased the muzzle velocity of balls. However it seems that to penetrate 'bulletproof' plate, people still had to resort to the simple tactic of just using a bigger gun.

I can't provide more detail I'm afraid because I've lent out the book I have on this subject - Weapons and Warfare in Renaissance Europe. This has some details on penetration and ranges of early weapons.