There's a lot of disagreement about the actual differences between the four elementals. Many theories have been presented attack rates, damage done, armor class, avoidance, rate of bash/kicking, spell damage, and so on. About the only thing that everyone agrees on is number of hit points (earth has most, followed by water, then air, then fire). Everything else is inconclusive, however, I'll try and present some of the most popular opinions here.
Earth: The earth elemental has the most hit points of any of the elementals, and also casts a root-type spell pretty frequently, that is pretty tough to resist. For these reasons, it's a popular choice for soloing. The popular belief (most empirical evidence seems to support this) is that he gets hit more often than the other three, due to low agility, but his extremely high HP make up for this. The root spell does not appear to do any damage, it's just a root. GZ said once that the earth pet had the highest strength, though all pets seem to have the same maximum damage, so the best guess is that the earth elemental hits for maximum damage more often than the other three. Tests of this have been inconclusive. Some people believe that the earth pet has a lower attack rate than other pets, but this has also not been shown conclusively.
Water: The water elemental has nearly as many hit points as the earth elemental. In addition, it's agility is better (it gets hit less), and it has a super fast regeneration. These factors also make it a popular soloing pet. It casts a direct damage spell, though how much damage this does in relation to its attack and the DD spell of fire is unknown. Its strength and attack rate (if they are indeed applicable and differ between the elementals) is probably in the middle, between earth and air.
Air: The air elemental is the opposite of earth. It has relatively few hit points, but a higher agility, so it gets hit less often. Generally, this isn't enough to make up for its lower hit points, so it won't last as long as an earth elemental, though. In addition, it casts a stun spell which, when successful, will always stun a caster. Thus, the air elemental is a popular choice when fighting casters, especially in a group, when the elemental won't be taking many of the hits. Many people say it also has a higher attack rate, but again, tests of this sort have been inconclusive. It also casts invisibility on itself, but since you can do that with any of your pets, it's not that worthwhile an addtion.
Fire: The fire elemental has the fewest hit points of any elemental, and only average agility (on par with water). Thus, it tends to die rather quickly. However, common belief is that it does the most damage of any elemental, through a combination of high attack rate/strength and direct damage spells. So, it's a popular choice in full groups, where it's not in danger of taking too many hits. Also, it casts fire shield on itself, which is almost always better than the equivalent shield you can cast, which means anything attacking the elemental will take significant damage. If you can keep him healed, most things will kill themselves eventually by beating on your elemental.