After much consideration, and after reading this thread, it occurred to me that I am the only one truly qualified to answer this question. You see, I am both a Jedi Master and have captained my own Constitution-class starship. Now, you are probably wondering, 'But Captain, how is it possible for you to have such important roles in two different universes?' Actually three, including this one. Elementary. I use my self-invented PUS Device. PUS stands for Parallel Universe Shifting. It allows me to shift from one universe to another, at will, having different identities in each. When used in conjunction with my TDD (Temporal Displacement Device), I can pretty much exist anywhere. Now I could explain how these devices operate and are used, but I believe that doing so would cause a catastrophic DP server crash, and make vauge cry. For more information on these devices, read everything ever written on quantum physics, temporal mechanics, and the collected works of Douglas Adams.
Moving right along. One must always remember that for a universe to be great and powerful (and having visited many, I would know), two important criterion must be explored: technology and characters/races.
Let's look at technology, first. When we look at lower level, or hand-to-hand/ground technology, both universes are fairly equal. Phasers or blasters? Pretty even. Star Wars may have a slight edge in mechanized artillery, but remember; in the Star Trek universe, there is far less need for ground combat, so the technology is less developed because it is less needed. When we discuss other, more 'civilized' hand-to-hand combat, Star Wars has the edge. Nothing Star Trek can offer can beat a lightsaber. A lightsaber can also neutralize a phaser, though I've never seen what it could do against a wide-angle disrupter. Interesting thought.
Looking at higher level/starship technology is more interesting. The top level Star Trek starship could take out an Imperial battlecruiser. It might take a couple of shots, but the fact that, with a concentrated effort, numerous tie-fighters could do it, is good evidence that the Enterprise (A-E) would be victorious. Seems to me that deflector shield technology is more advanced in Star Trek. Furthermore, Romulian plasma weapontry would also cause severe damage, though Romulan ships are more suseptable to being damaged when coming out of being cloaked.
This then brings us to The Death Star. How could Star Trek defeat it? Well a look at the construction of the Death Star is important for this analysis. The Death Star is at it's most powerful when it is anchored to a friendly planet, something that severely limits it's ability to be an aggressor. The Death Star II's deflector/defense system was tied to the Endor moon. Without this, it's defenses were no better than the first Death Star's. Though the sheer size of the vessel, it's amount of guns, and it's decent defense system, would make it difficult for even several Star Trek vessels to destroy, a simple solution would be to fire a Genesis Device into the Death Star. 10 minutes later, instead of a near impenetrable battle station, you'd have a planet sprouting daisies. Not very scary.
Sorry, with this rather simple solution, Star Wars loses, though only slightly.
Let's now examine characters/races. A common misnomer is the power of the Jedi mind trick. Though a powerful technique, it doesn't work on the strong-minded. It would never work on Kirk. Powerful as they may be, The Q, despicable as they are, could just think the Jedi out of existence, though it is conceivable that a Jedi as powerful as Yoda (or myself) could use The Force to manipulate midichlorians in such a way as to negate The Q's power. Unlikely but possible. But beyond that, what would the Jedi do with pure energy creatures like the Organians or The Founders? Can't do a force-push, and force-lightning would be useless. It stands to reason that creatures like the Organians can manipulate The Force itself, something even the Jedi can't do. Furthermore, when it comes to standard hand-to-hand combat, who do you want on your side, Clones/droids or Klingons/The Borg/The Dominion? I'd bet on a Klingon, for example, any day.
So, after this rather scholarly report, I must conclude that in a battle of universes, though the 'fan-favorite' would be Star Wars, the winner would be Star Trek. Doesn't mean I'm giving up my lightsaber anytime soon. I can have the best of both worlds.