Though this is from the Quik-Pak trailer website (gonna be biased toward 2 wheel), I own both types and generally find this info correct:
One -wheel trailers:
The one-wheel trailers definitely have the advantage on single-track trails. A narrower trailer is less likely to clip a tree or catch on a shrub. The center of gravity is lower than panniers, making it easier to control the load than it is with a higher pannier setup. With the current one-wheel trailers, low ground clearance may be a problem on uneven surfaces or riding over an obstacle. Placing all the weight on one trailer wheel and the bike means that the bike carries at least one half of the weight. Since a one wheel trailer does not balance the load on its own, the rider must always balance both the bike and the load; making it more difficult to lift the bike and the trailer from a resting position, or to move the bike and trailer while stopped. One-wheel trailers also increase the effective length of the bicycle, creating a machine with a longer wheelbase. This may make the bicycle/ trailer combination difficult to control. Quick maneuvers, crosswinds, or an unstable bike can create high-speed shimmy and make the bike impossible to control.
Trailer and bicycle frame stress can also be an issue with one-wheel trailers. The side-to-side motion created while riding the bike is transferred directly to the frame of the trailer and bicycle; sometimes causing frame struts to break from the metal fatigue created while riding with a heavily loaded trailer.
Hooking up a one wheel trailer is a job for a three-handed individual. Balancing the bike while attaching both sides of the trailer hitch and slipping those little pins into place can be a challenge. It helps to have a strong friend or the perfect wall to support the bike. Heaven forbid that you tour alone or in an area where a handy leaning wall isn't available.
When traveling with the one-wheeled trailer it is difficult to fold and carry, or to store them as luggage. Walking with any one-wheeled trailer once again requires both hands to balance the load and move the trailer. At camp it is simpler to just leave the trailer in one place and tote your gear to and from your tent.
Two-wheel trailers:
A good two wheel trailer can be quickly attached to, or unattached from , the bicycle by any two handed person. Just hold the bike with one hand and pull the self-balanced trailer up to, or away from, the hitch mount and slip it on or off the ball. A two wheel trailer can be easily moved about camp with one hand. With two wheels balancing the weight, the rider never has to lift the weight of the bike and the trailer as a unit. When necessary it is easy to simply drop the tongue without danger of tipping or spilling the load.
Two wheeled trailers may widen the effective width of the bike slightly, however most trailers are about the same as the handlebar’s width and rarely create a problem (you may not be able to store your unfolded trailer in the bathroom). The tongue weight on a two-wheel trailer rarely exceeds one third of the load weight, reducing the adverse effects created by your load under extreme situations. Any weight supported by the bike is supported at, or in front of bike's axle, creating no increase in effective bicycle length.
http://www.quik-pak.com/Bike%20trail...20or%20two.htm