In general, crossbow archers are much easier to train and can be easily mass produced by a sophisticated ancient military industry, such as that of Chinese Qin empire and Roman empire. The manufacturing could be broken down into the fashion of a production line. Each piece can be manufactured separately with ordinary materials. In fact, remains of Qin dynasty (around 500-200BC) crossbows and arrows show manufacturing precision greater than 0.1mm, in different sites throughout the empire, distancing thousands of kilometers apart. On top of that, it doesn't require exotic materials like those used in curved bows. Ordinary bronze and iron are used for the mechanisms and average high quality wood for the bow.
Far East composite bows require tons of specialized materials (up to more than 100 for Hun style) and are very time consuming just to prepare the raw materials (2 years to dry the hide). It is the same story with the famous English longbow (a year or two to dry the wood). All traditional curved bows require very skilled craftsman and years to produce, whereas Qin counterpart only have to teach each guy how to make one part of the whole crossbow. Hence they are very limited in production.
On top of that, traditional bow archers are very hard to train. Most of these men start at early ages and utilize those skills throughout their lifetime. Unless you are in a nomad group, it is very hard to train a dedicated military archer. Training a professional archer with no previous experience would take years and cost a lot of resources. On the contrary, crossbow archers only have to learn how to pull the string and cock it, then how to aim and pull the trigger. They can be considered adequate for battle with as little training as just short few days.
But traditional curved bows have landslide advantage over crossbows in terms of firing rate, flexibility, and accuracy. Anyone with decent knowledge of archery would understand this. Well trained archers can fire at a very high rate, even close to 15 to 20 a minute in a short burst. The experience of a curved bow archer allow them to hit targets with much better accuracy, even behind obstacles since these arrows tend to travel in a curved trajectory. They can also shoot target from much trickier angles. The greatest advantage of a dedicated archer is that they have much better accuracy at greater distance. The intimacy between the archer's hands and arms with the bow allow them to have much better feel of the weapon.
So to conclude the comparison, they both are very powerful weapons, but in different manners. Traditional curved bows are more for elite units, more expensive and take long time to both train the men and to build the bows. They are like the marksmen and machine gun of their time, both expensive and effective. Crossbows are easy to mass produce and can easily be mastered.
There is really no winner or loser. Both style helped armies defeat their enemies. The Qin empire conquered all of China partly due to its fearsome crossbows. They were mass produced with very high quality and precision, similar to today's production line. The English showed a very different approach. They didn't have that much population and the number of longbow men even more scarce. But despite of small number, they were still very deadly and easily massacred the heavily armored Italian mercenary cavalries.
