What are the differences in aircraft performance between tractor and pusher propeller designs?
Well one major difference is the blade pitch, pusher props will seem to be backwards. 

You must also modify the engine's bearings to be able to support the propeller mounting and crankshaft in compression, rather than tension.

A pusher prop's engine also has to be cooled differently than a tractor, because the propeller is not blowing air over the engine. Therefore some ducting and ram air magic has to be done, or some other system of cooling implemented. Many pusher-type planes have had severe issues with cooling.

Note: some of the next few might not necessarily apply to pusher-type planes where the engine is mounted above the fuselage, like the Lake Amphibian and many other seaplanes, or some other oddities like the B-36. I am talking about more conventional pushers like the Velocity, Beech Starship, or Cessna 337.

Another issue is FOD. A pusher prop will likely be mounted behind at least one of the wheels, which can kick dirt and rocks up into the propeller. Now imagine a prop that is very close to one of the mains. Even on paved runways you encounter rocks and debris occasionally, and it only takes one hit in the right place to produce a failure-causing nick.

Compounding on that last one, when a tractor-type plane takes off, beacause the prop is forward of the center of mass, the propeller will rotate up, away from the ground. But a pusher prop is usually just the opposite: it will rotate down a bit, increasing the chances of a prop-strike. This is one that is heavily considered and worked-around while building pusher-type planes, but while you can mitigate the danger by placing the prop closer to the CM, or making the landing gear taller, you can't completely remove it as far as I know.