It's the argument that never ends but in all fairness, it is a good one: Star Trek vs Star Wars, something almost guaranteed to divide your friendships, end marriages, force choices between the Dark Side and the Prime Directive.
There are some problems with the arguments, however, and they tend to boil down to what is cannon what is not. Why do Star Destroyers blow apart asteroids in a single shot but also only just manage to damage an R2 unit? Why do the Federations' anti-matter weapons sometimes destroy a ship in one shot and other times require whole salvos to scratch the surface? We do our best to explain these things away but when it comes down to it, the inconsistencies are there to make good storytelling, to further the plot, and to aid in nail-biting narrative. Fair enough, you might say, but at the end of the day, this makes for some infuriating debate.
In light of this we are going to do something a bit different; instead of focusing on the weaponry (though it, of course, will be mentioned) we will look at the two sides as a whole. I will attempt to break down these incredibly rich universes into smaller categories, and from there we will examine both sides of each issue. A war is more than what is simply on the field of battle.
It's safe to say the Galactic Empire is very much a superpower and has massive industrial strength. It is capable of the creation of megastructures such as the Superstar Destroyer and the Death Star... more than once. It also seems that they can build these projects on relatively fast time scales. The Empire seems obsessed with creating said megastructures... much to its detriment. Zachary Feinstein, PhD, did a study on the projected cost of each Death Star to the galactic economy and concluded that on the destruction of such a station the Empire would need a bailout of at least 15% to 20% of the Galactic GDP. Having said that the Death Star is a superweapon and puts the Empire at quite an advantage... for now.
The United Federation of Planet's economics work in a very different way than that of the Empire. While it does need to do some hard mining for certain elements such as Dilithium, most of its needs are provided for by one invention; the Replicator, a device that can rearrange molecules in almost any combination in order to produce almost anything, seemingly out of thin air. While the Empire has many industry-focused worlds, the UFP simply does not need them. This makes production faster and more precise for the UFP, providing them with a huge advantage in the event of a struggle with the Galactic Empire. This also makes the Economy of the UFP less vulnerable in a conflict, with Starfleet only needing to provide a defence to a few key locations. The Empire, on the other hand, would need to secure far more facilities, having a more traditional style of industry that would be more vulnerable to damage or destruction.
For this, I have to give the United Federation of Planets the economic victory.
UFP 1 GE 0
The Galactic Empire, like the Federation, is able to go faster than light. In the narrative of Star Wars, it is often referred to as light speed but it's clearly much faster than that. We see them hopping from system to system in very little time, even managing to move structures such as the Death Star. In Star Trek, we see a similar mechanism, but I think its safe to say it takes much longer.
In Star Trek Voyager, we see the USS Voyager trapped at the far end of the galaxy facing a journey home that will take decades. While an argument could be made that technology exists in Star Trek to make this journey much faster, such as the Borg transwarp conduits or the Caretaker's array, this is not a technology that is readily at the UFPs disposal. While the Federation can travel across the stars at an impressive speed, it has nothing on the Empire; towards the end of The Empire Strikes Back, we see the rebel fleet hiding out beyond the edge of the galaxy. Even with the Federation's warp 9 engines, they would never dream of making a journey like that.
The Empire can clearly outflank the Federation, putting the UFP at a distinct disadvantage here.
UFP 1 GE 1
The Empire seems pretty bad at this. One of my favourite scenes in Star Wars is the Millennium Falcon hiding from a star destroyer by attaching itself to the back of its command section and then drifting away with the garbage, something the Falcon would not have a hope in hell of pulling on a Starfleet vessel. Starfleet have sensors that can detect sub-atomic particles, let alone a ship, and can do this from huge ranges. Starfleet enemies need to invent very creative ways of bypassing the detection nets or their patrolling ships. The Falcon is primitive compared to Starfleet vessels and it can evade detection from the Empires top of the range warships—this alone is enough to give this one to the Federation.
UFP2 GE 1
This category is not as simple as it seems.
The Empire certainly has the upper hand when it comes to sheer numbers and variety of equipment, but the Federation has it in technology and, in most cases, the tech will come out on top... but not always.
If the Empire manages to break through Starfleet lines and invade a Federation world, they could likely do so in huge numbers and with numerous fighting machines. The Federation would find this hard to combat by traditional means and may need to resort to guerrilla tactics. Guerrilla tactics, however, are what the Federation ground forces would excel at; where the Empire is fast in space the Federation holds the upper hand on the ground, able to teleport directly into the heart of enemy strong points with personal or weapon systems, undetected and unopposed. It is possible that the Empire could succeed with an invasion force, but in the long run, it's probably that they would have trouble maintaining an occupation.The Federation would struggle to invade Empire held worlds but could foster resistance movements against the Empire on their own worlds. It seems that most Empire held words are not exactly willing participants within the Galactic Empire and could theoretically jump at the chance for rebellion.
All in all, I would say this is a draw and one that leads us to a much darker problem.
UFP 3 GE 2
The Galactic Empire, frustrated by its lack of significant progress in its war with the United Federation of Planets, may fall back on its superweapon. There is a real possibility that if it can't take and hold a planet then it will simply resort to destroying it. That being said, bringing the Death Star into play is risky for the Empire, as we discussed before; if it were to lose this asset it would be a huge blow to the economic stability of the Empire. On the other hand, it could be the turning point for the war and bring the UFP to its knees.
Now we need to look closely at the attitudes of each side. Let's say from the Empire's greater ability to move through space that the Empire can strike at least once with a massive surprise attack. The Federation will be able to detect the Death Star and its escort fleet from far away but will, in all likelihood, be too late to effectively counteract the Empire's plans. Before Starfleet has the chance to react, Vulken has been destroyed just like in Star Trek 2009. What does the Federation do in reaction to this? Do they surrender to avoid further atrocity? It is doubtful. When the Klingons hit Axonar, the Federation mobilised and so began the Klingon war, a war that the Federation may have easily lost. When the Borg launched their attacks in "Best of Both Worlds" and "First Contact," the Federation stood firm, and when the Dominion came it did so again.
While the Federation has no stomach for conquest, it does have a tendency to be unceasing in matters of self-defence: The hypothetical act of the Empire destroying Vulken will spread fear through the Federation but not foster defeatism.
The Federation will do everything to engage the Death Star and when it does it is highly likely that the fleet's sensors will quickly find her flaws, flaws we now know where purposely put into its design. This is what happens when you have forced labour rather than patriotic labour! Starfleet will quickly exploit this defect, just like the rebellion did, but this time they will not need to rely on a lucky shot from a fighter; they can beam a torpedo right into the reactor. Boom.
The Death Star, the Empire's superweapon, a megastructure that has come close to crippling the economy of the Empire, will be dealt with swiftly and mercilessly. This act will change the face of the war.
The Federation does not, however, have the capability to "conquer" the Empire, but that doesn't mean it won't stop it in its tracks.
UFP 4 GE 2
Who would win in the long run? I can't really say, but the Federation continues to advance in technology whereas the Empire is somewhat stagnant, or at least in comparison.
If I were to speculate, however, I imagine that the Federation would push for peace at every opportunity. The Empire would be struggling with its weakened power internally, and the economic crisis that the war has delivered will be crippling.
Both would survive but only one would live long and prosper.
Peace (and long life).