The common measure of the power of a bow is its draw weight. The draw weight is the force (expressed in units of weight), required to draw the string to its maximum position. This measure gives a good indication of the raw power of the weapon, but there are a lot of other factors that govern its effectiveness.
Draw length is the distance that the string travels from its release position to its maximum draw position.
After release, the projectile accelerates through the draw length, and as soon as it leaves the bow it begins to decelerate.
However, because the bow is essentially a type of leaf spring, the energy stored in the bow is transferred to the projectile as the bow straightens, but as the bow straightens so less power is delivered, so acceleration peaks very shortly after release, and decreases until the projectile leaves the bow.
Arrow length must be longer than or equal to draw length in a simple (straight) bow because the shaft must be guided by the archer's hands. In a crossbow it can be shorter, since the stock guides the projectile.
A longer arrow will be heavier. More projectile weight means more inertia to overcome in acceleration. Longer simple bows require longer arrows, so to minimize weight, the arrow shaft must be lighter, and to be lighter it must be thinner. A light shaft is flexible. On release, the force of the bow against the inertia of the arrow compresses the shaft, causing it to bend. A fine, even shaft will form a balanced double bend, known as snaking. The longer the shaft, the more likely there will be grain imperfections that will cause unbalanced snaking, and an inaccurate arrow. Modern fibre arrows flex very little, since they are extremely light and extremely uniform.
A longer arrow will, however, be more accurate at longer distances. The long shaft will stabilize it. This means that the fletching (or flight) can be smaller, which reduces drag, and further increases accuracy (or rather, limits the effects of flaws in the fletching on accuracy).
In the crossbow's shorter draw length, a shorter arrow can be used. So a short fat arrow (usually known as a bolt) can be used with the same draw weight, and it won't flex. This has the advantage that projectile stock quality need not be quite so exacting. However, it has none of the stabilizing effect of the long, thin shaft, and so it needs a relatively large fletching, which causes drag.
These factors combine to make the crossbow much more accurate over an average number of shots at short distance, but accuracy decreases very fast with range. At shorter distances (under 100yds), a crossbow is slower but quality of manufacture of the projectiles has only a small effect on accuracy. At longer distances, the simple bow still requires the arrows to be of high quality to achieve accuracy, but the accuracy it can achieve is much greater.
The bow has two uses as a tool.
One is for hunting. The crossbow us next to useless for hunting as it is too noisy at close range and inaccurate at long range. The simple bow is almost silent.
The other use is of course for killing people who are (presumably) trying to kill you. Military historians and theorists, as well as people who can do a little simple math, will tell you that in projectile weapons, two factors trump everything else. The lesser of the two is cover: can the weapon be loaded and fired from cover. Both actions are easier for the simple bow; the crossbow is hampered by cover, but slow to reload, so cover is more important to the crossbowman. The greater of the two is rate of fire.
In rate of fire, the crossbow suffers the most terrible of compromizes. The only way to maximise rate of fire is to reduce loading time, and that can only be done by sacrificing mechanical advantage, and in consequence, draw weight. To summarize: a crossbow can be fast to reload or it can be powerful. It cannot be both. Even the fastest loading crossbow (the simple lever crossbow) has a rate of fire that is at best half the rate of the simple bow.
The major advantages of the crossbow are in fact economic. Manufacture of crossbows is a little more expensive than manufacture of simple bows, but only because the best simple bows require considerable expert skill and practice to manufacture, whereas the crossbow is complex. However, crossbow ammunition need not be of a high quality, so the projectiles for crossbows are much cheaper to manufacture than accurate arrows for longbows. All this pales into insignificance, however, when compared to the cost of training the soldiers.
It takes a few weeks at most to train someone to use a crossbow effectively on the battlefield. However it takes many years of regular practice to become proficient even in short simple bows with a relatively low draw weight.
And so to Skyrim.
Skyrim is a game, and it is a game about being a hero (or the opposite). It isn't a game about being a battlefield soldier, or a medieval general. The hero is, one assumes, something of an expert at arms and weapons, and has both the time and inclination to practice enough to perfect his skills.
Crossbows were added with the Dawnguard DLC (downloadable content—an extension to an existing game that is downloaded from the internet, M'Lud).
The crossbows in Skyrim are short, simple lever crossbows. Even when cocked by a massively muscle-bound Nord Hero, the draw weight must be pitiful, given the rate of fire that you can achieve. Yet they seem to be more effective than simple bows. This, of course, is the physics of the rule of cool. Crossbows are cool (apparently) so Skyrim has to have crossbows, and they have to be really really effective.
Personally, I don't think crossbows are cool. The crossbow is the weapon of poorly trained cannon fodder, not the weapon of a hero. I get that as a means of putting a stake through the heart of a vampire it makes a certain poetic sense - though in what way a conventional wooden arrow isn't a stake, I've yet to understand.
Next there will be pistol crossbows. I wish there were an html markup for massive, withering disdain. If there were, I'd have written that: <massive, withering disdan>pistol crossbows</massive, withering disdain>. Probably in comic sans. Being hit by a bolt from a pistol crossbow is rather like being poked with a damp twig.
Sigh. I like Skyrim. It has a great deal of awesome, and an awful lot of silliness. Don't even get me started on the dungeons full of useful gear and all the heal and buff potions left lying around just before the difficult bits.
Probably time to call it a day.
Were you wondering where the spurious/tenuous connexion to writing was? Here is your Aesop:
There is no substitute for research. Because while you can have some hero-dude who has a special skill that means he can both fabricate and use crossbows that are as effective or even more effective than simple bows, you should not do so in ignorance. The Rule of Cool is at your disposal, if applied sparingly, but you should do so knowing that you are doing so.