Star Trek Vs Star Wars  

I am a sci fi fanatic and a die hard trekkie. I will read or watch just about anything in that genre except horror and demonistic tales that they try to classify as sci fi. That being said, I have to choose the federation over the empire. The federation used more logic and honor than the empire did. In the end logic will get you every time.

This battle is very old. Star Wars is a warlike series with more focus on, well, war. However, if weapons and shields are equal in strength, than my bet is the Federation will win.
The Federation's phasers can go through a variety of settings. Instead of simply used to kill other people, they can be modified to stun or vaporize. They can cut a hole through a wall, or destroy an entire building. Star Wars has the force, but they won't have a chance to use it. Star Trek will usually do their battle on board a starship. Unless a force choke can be used across space and past an energy shield, they're at a loss.
Star Wars has their Death Star. Yes, they can destroy an entire planet, but the Federation fleet has destroyed a Borg Cube before, which is much more deadly due to it's regenerative hull and adaptive abilities.
As I recall, photon torpedoes can be modified to increase their destructive yield. They can be powerful enough to destroy a small world. The Empire has no chance over Star Trek's awesome power.

OOOOOOOOH, now THAT'S a real thinker, hmm...
Well you have to understand that the Empire is, of course, evil, and would have to lose in the end. Therefore I would say that the Federation would win one on one. But then you could have one of those twisted endings where something totally different happens, like the two merging into one alliance, or both of them destroying each other and some new organization coming out of the dust. Tough question.

Star Trek seems to be more up to date than star wars was. Star Wars was just about wars where Star Trek was about exploring outer space and getting to know about new races.