The descriptions of each of the different bow types can be seen
here
As can be seen from the table on that page, each bow type has a different combination of range, attack speed and flight arc (the path the arrow takes and a good measure of accuracy). This gives each bow type a set of characteristics that make them useful in different situations.
The longbow is a fairly reliable bow. It has a long range and reasonable refire rate, along with medium to good accuracy. This doesn't make it the best bow for damage, but it does allow you to deal decent damage at a good range and is exceptional for pulling groups of enemies in PvE.
The shortbow has a fast refire rate, making it good for dealing damage. It is also fairly accurate, although it has the shortest range.
The flatbow has a fast refire rate, making it as good for damage as the shortbow. The flatbow also has a longer range than the shortbow. However, the flatbow's larger flight arc makes it less accurate, though this may prove to be an advantage in some areas as it enables you to fire arrows over most obstructions that might be on the terrain.
The recurve bow has a very low flight arc, that reduces flight time and so increases accuracy considerably; this makes it the ideal bow if you want to focus on interrupts. Its average refire time doesn't make it the best bow for damage, however.
The hornbow has the slowest refire rate, this may seem like it isn't the best bow for damage, but its flight arc gives it reasonably good accuracy and has an inherent 10% armor penetration that is applied to each arrow you fire. I haven't done the calculations to compare the effect of its armor penetration on its average DPS, but my bet is that this would be a reasonably good bow for dealing with high level monsters that have high armor.