Not to stir things up too much, but I thought I would provide a counter view to the argument that crossbows were more accurate. This is an excerpt from Gunpowder and Galleys, by John P. Guilmartin, which is difficult to find, although the chapter on weaponry is available on the author's website and can be found here:
http://www.angelfire.com/ga4/guilmar...m/Weapons.html

Having first been published in 1974 it is certainly a bit dated by now (although there was an updated version published in 2005 -- I haven't been able to get a copy of it). Most of the information references Payne-Galloway, but nearly everybody does -- a hundred years ago they got away with doing things that couldn't be done today; like going into museums and firing their 400 year old crossbows!

However, this is, so far, the most technical analysis of crossbow accuracy that I've read, so I wanted to present it --

The reasons for the crossbow’s inaccuracy are somewhat involved. They begin with the mechanics of the release mechanism (see Fig. 4). Where an archer, by precisely controlling his release, could ensure that the energy in his bowstring was smoothly transmitted to the arrow, the crossbow release mechanism released the cord abruptly and somewhat erratically. Instead of being smoothly accelerated in a carefully controlled direction, the crossbow bolt began its voyage lying loosely in its trough, and was then ‘slapped’ into flight with enormous force. Crossbow bolts had to be made short and thick with a flat base in order to prevent the tremendous impact of the cord from reducing them to splinters.12 In view of the need for strength and the basic inaccuracy of the crossbow, war bolts were often very crudely made, having a single leather fin set into a slot sawed across the base of the bolt. The aerodynamic inefficiency of the resultant shape sharply increased drag and therefore reduced the maximum range. This was aggravated by the considerable and unpredictable vibration which the impact of the cord imparted to the bolt. By further and inconsistently increasing the aerodynamic drag of the bolt this vibration additionally reduced both range and accuracy.
I am not claiming that this is necessarily the case, just that it seems fairly convincing on the surface. If Guilmartin is correct, then why do historical reports make the opposite claim about crossbows versus bows? I'm willing to speculate a little:

1. Perhaps the authors weren't as trained in the use of bow as they needed to be? It is generally agreed that learning to use a bow effectively was basically a life-style, something begun at childhood. On the other hand, the crossbow was something that could be trained in a comparatively short amount of time. If your "example" bowman was lacking the background of the former, then we could expect a crossbowman to out perform him.

2. An agenda, or embellishment, or both. Most writers from that era don't seem to have been too objective (even if they wrote as if they were).

3. A practical assessment, instead of a measure of the intrinsic capabilities of the weapon. We can tie this point with the other two in a very logical way --> if I give three weeks of bow training (point 1) to one soldier, and three weeks of crossbow training to another: the crossbowman will be more accurate with his weapon, therefore crossbows are more accurate than bows (point 2). ;-) This isn't necessarily a deliberate attempt to misinform, it's a practical statement, and in my opinion it's very important. But it can provide confusion if we are interested in the intrinsic potential of the weapon.

Or perhaps Guilmartin's analysis overloooked some counterbalancing factors? Or maybe he was just wrong? :-) I don't know, but he writes pretty convincingly!

@Incanur -- you mentioned a Scientific American article from 1985, where they conducted drag tests on actual crossbow bolts. Unfortunately, I do not have access to that magazine. As that would be more recent, and probably pretty scientific, testing, I would love hear more about it. All I could find out about it by poking around online, was that they tested ancient Roman crossbow bolts. My understanding is that most Roman "crossbows" were largish siege weapons? And such a bolt might have had considerably different dimensions from a medieval one?