Different tools for different uses.
A crossbow is not necessarily better than a bow in all circumstances, and the same vice-versa. The two weapons have different capabilities which make them more or less desirable in certain scenarios.
Let's compare the two and see how they stack up:
First off, both crossbows and "traditional" bows both have the same general function: to use a mechanical spring to throw a stick with a metal tip at a target with enough force to penetrate the target at range. With a bow, the shooter holds the bow with one hand and draws back the string with the other hand, relying on muscle power to hold the draw. With a crossbow, the drawn string (drawing a crossbow is called "spanning," by the way) is held in place by a catch, usually a rotating nut or a retracting pin. Depressing a lever or pulling a trigger releases the catch. This means an archer with a crossbow can span his or her weapon and hold it at the ready for an indefinite amount of time before shooting while an archer with a bow must constantly exert himself to hold the weapon at a draw.
This would make it seem that the crossbow is the easier weapon to use, and in a way it is. Shooting a bow, especially one that is powerful enough for hunting or warfare, requires lots of practice to build up the necessary muscles and develop muscle memory. A crossbow, by contrast, can be spanned and shot by pretty much anyone.
Incidentally, a crossbow’s accuracy, deadliness, and ease of use earned it the ire of the church in the Middle Ages. It is said that both Popes Urban II (in the 11th century) and Innocent II (in the 12th century) banned the use of crossbows against Christians (though using them against non-Christians was A-ok). (Although, Innocent II’s Second Lateran Council also banned bows and slings for use against Christians, but everyone only ever seems to remember the crossbow part…)
So, if crossbows are so awesome—allowing for a relatively untrained peasant to kill a knight or (as Richard the Lionheart found out) a king—why didn’t they totally supplant bows? In some medieval continental armies, they pretty much did.
As a case study, let’s take a look at the Battle of Crecy, fought in August of 1346 between the armies of Edward III of England and Philip VI of France.
King Ed brought with him on his campaign around 5,000 archers armed with longbows. Now, as bows go, a medieval English Longbow is almost as simple as you can get. It is very elegantly and carefully designed and shaped by a skilled craftsman, but it is, at the end of the day, a well-shaped stick, and compared to a crossbow, dirt cheap. Edward did not provide the bows for his army. The men were required to bring their own, and were paid a few pennies per day. It is also worth noting that the English at this time had a whole longbow culture. Lots and lots of freemen were compelled to own a bow and practice with it frequently.
Philip, on the other hand, had hired a mercenary company of elite Genoese crossbowmen. Northern Italian crossbowmen, especially the Genoese, had a reputation of being the best. Some estimates put their number at around 6,000, probably less (though Froissart’s chronicle claims 15,000). (as an aside, not counting the Genoese, it's possible, based on some estimates, that there may have been at this time only about 2,000 trained crossbowmen available to muster with the army in all of France).
After being harried by the French army, Edward barely managed to get his army across the Somme and—knowing he wouldn’t be able to evade his pursuers any longer—decided to make a stand on a slope, with his left flank secured by the River Maye and the town of Crecy, and his right by the town of Wadicourt. The French billeted in the villages to the south and east, confident of victory.
The French plan called for the Genoese mercenaries to deploy across the front of the army and use their crossbows to create a screen behind which the French cavalry could form up and prepare for a charge. In a rush to get things underway, the Genoese were forced forward under protest before they had a chance to properly prepare and rest up, or even retrieve their pavises (shields to hide behind while they spanned their crossbows) from the baggage train. It is possible that a sudden rainstorm swept in shortly before this, and the wet weather damaged the bows. Unstringing, restringing, and keeping the animal sinew strings dry was easy for the longbowmen (even today, any archer will tell you that pretty much any bow except a compound bow is never kept strung except when you’re actively using it). Unstringing and restringing a heavy-poundage crossbow is not quite as straight forward.
The end result? According to Froissart, the crossbowmen let loose a volley, which fell short of the English line. The English archers took a step forward, bent into their bows, and released volley after volley. The longbowmen were able to shoot faster and farther then the crossbowmen. The Genoese broke and ran, running smack into the advancing French. In a fit of disgust, someone among the French, maybe King Philip, maybe the Count of Alencon, maybe both, ordered the fleeing Genoese to be ridden down and killed.
It is tempting to put the range and accuracy failure of the Genoese crossbows at Crecy on the wet weather, but that’s not entirely the whole story. A projectile launched from either a bow or a crossbow must be capable of withstanding the forces put upon it when shot. This necessitates a certain diameter for a given length. A crossbow, because of its construction, can have a very high draw weight (how much force is required to draw back the string), sometimes upwards of 1,000 lbs. A bow, by contrast, cannot be too heavy, or no one would be able to draw it. The warbows recovered from the wreck of the Mary Rose are estimated to have draw weights upwards of 100 lbs, and were intended for use by professional archers. Ok, so what? Well, we also have to accommodate draw length. A bow has a draw length of about 2-2 ½ feet or so, depending on the archer. A crossbow has a draw length of maybe a foot at most. The longer the draw length, the more time the spring (for that’s all the limbs of a bow or crossbow are, springs) has to impart its energy to the projectile. As a result, a crossbow requires a higher poundage to get the same velocity out of its draw length that a bow does.
Also, because of the draw length, a bow shoots an arrow approximately 3 feet in length. This arrow is long, skinny, and stabilized by three feathers (called fletchings) at the tail end. A crossbow shoots a bolt (also called a quarrel) maybe a foot long, necessarily thick to withstand the higher forces, and usually stabilized by two simple fletchings (in the Middle Ages, parchment or thin leather was frequently used). Three fletchings are possible, but relatively rare until more modern times, as it’s kind of tricky to get three fletchings to work with the groove you have to set the bolt in.
When it comes to stability in flight, long and skinny beats short and stubby. This means that at longer ranges, a crossbow bolt will start to tumble and lose velocity, meaning you have to get even more poundage to reach out and touch someone.
So, let’s take score:
Bow. Pros: cheap, simple to construct, accurate at range, faster “rate of fire”. Cons: requires extensive practice and training, practical limit on poundage.
Crossbow. Pros: Easy to use, powerful punch, can hold a draw for long periods. Cons: more difficult to make and maintain, stubby projectile loses punch at range, painfully slow to reload, especially for heavier poundage.
Sitting in a castle with all the time in the world to snipe that enemy knight? Grab a crossbow. Need to punch through a knight’s armor and don’t have decades to train? Grab a crossbow. Need projectiles down field quickly or on the cheap? Grab a bow. Need something simple and easy to maintain to keep your family fed with game? Grab a bow.