Enterprise vs Millenium Falcon?

Recently Io9 posted a youtube video where this question was posted to various celebrities attending the San Diego ComiCon (link to article here). The answers for the most part are fairly poor, which is surprising given that these are all people in SF. The exception is Adam Savage who does know his SF tech tries to argue from a visual history of the respective franchises, and Cas Anvar who knows how characters drive plots: "Han Solo has got way too much luck."

Star Trek vs. Star Wars?

The whole area has been filled with many pages of acrid forum threads and loud discussions at con panels (in good fun of course). Perhaps the most comprehensive (to be kind) collection of pages on the matter are those written by Michael Wong at StarDestroyer.net. The general tone of the entries shows a strong bias  for the SW empire over the ST federation. On the other hand, there are many ways in which it is obvious that the Star Wars universe from the point of view of weapons and warfare is immeasurably superior to the Star Trek's Federation. Of course this can get ardent trekkies in a twist, but to be fair it is tied to the very nature of the universes presented. Staying away from books and "technical" manuals, and only sticking to what has actually been put to film and video, the evidence is there.

Domain

The Star Wars universe encompasses an entire galaxy. The Federation and immediate boundary domains (Klingons, Romulans, et.al.) only encompass a section of the galaxy. A sizable section, but only a section nonetheless. By the very admission of its creator, Gene Rodenberry, the vast volume of the Federation territory remains largely unexplored. In fact it is the very reason the exploratory mission of Starfleet exists. The Star Wars galaxy on the other hand is vastly populated by diverse cultures interconnected by a vast commercial network and controlled by a strong centralized bureaucracy. While they are worlds that may be considered "backwater" planets and planets about which little is known but not totally unknown. In, ESB, Luke is told to go to the Dagobah system, and that seems to be information enough to get there, in spite of being a theoretically uninhabited planet whose sole resident picked a place to literally get lost in.

Travel Capacity

The Star Wars universe allows travel via hyperspace pretty much from any part in the galaxy to any other part within a day. The Star Trek universe, and specifically the TV Star Trek universe requires long periods, of days and weeks to travel from one extreme of its territory to another, which are only sub-sections of the federation volume. Obviously Star Wars ship command an all important power in any battle, the ability to engage and disengage at any point at will.
     Some of the video commentators mentioned the difference in maneuverability of the Enterprise vs. that of the Millenium Falcon. On the other hand, the main ships of the Star Wars universe wallow about space like puffy white clouds rather than battlecruisers. In some of the newer, updated effect sequences of even the old show, the Enterprise and other Constitution class vessels are shown to be able to maneuver and shoot quite well, much better than any Star Destroyer at any rate. On the other hand, the Star Wars universe can count on swarms of highly maneuverable fighters than can stand up to close quarters with a Star Destroyer.

Offensive and Defensive capacity

This is the one place where I would consider handing the advantage to the Star Trek universe with one exception. Star Wars weapons are shown to create explosive  heating and electrical disturbances (arcing) on most targets they hit. The energy delivered can be substantial. At no point, however, is a Star Wars blaster or laser from either armed personnel or ship weaponry shown to actually destabilize the actual matter of the target; i.e. make them just disappear. This is, as many a redshirt discovered, a somewhat everyday thing in the Star Trek universe. The energies involved represent orders of magnitudes in difference.
     The one exception in this Federation superiority would have to be in the destructive force of the Death Star. While it is admitted that it would take "thousands of ships with more power [than currently known]" to carry this out in that universe, it nevertheless does manage to boil away an entire earth-like planet within a few seconds. That is really huge.
     Defensively, is a harder call. Both universes have shields. The Star Wars universe seems to allow larger ships to be able to sustain direct hits from their weaponry with little serious effect, other than heavy shaking, as long as the shields are up. Similarly, Star Trek shields seem to provide similar support. Here I would tend to call it a draw.

Sheer numbers

The Star Wars universe is an entire galaxy, and the main nemesis, the Empire, can count on a huge number of ships to bring to bear in a military situation. The Federation, while large, particularly in it's last shown "Next Generation" phase, is not as large as that shown or implied by the Star Wars universe. Of course the problem here is that for the Star Wars universe, we are not explicitly told what that size may be, but to realistically control an entire galaxy full of sentient populated planets (see Domain), it would really have to be big, if spread out.

 But how powerful are they really?

And this is where it all breaks down. You read up on StarDestroyer.net's calculations and citations and one comes up with yields in the hundreds, thousands, or millions of Gigawatts of output on their laser blasts. In spite of this, rarely is the actual damage caused by a hit from these weapons on ships (innocent asteroids notwithstanding) show damage in this range, the kind of damage that should vaporize small fighter craft and boil away huge holes through the entirety of a ship. Even the starship Enterprise rarely boils away asteroids or generates small novas with their photon torpedoes which are described as micro-antimatter  bombs (1g of antimatter with equivalent of matter or 2 thousands of a Kg mass should yield about 180 megajoules or 43 kilotons  or nearly 3 Hiroshimas).
     Then again the damage shown has nothing really to do with the theoretical or purported yields of the weapons involved. The key differences between the two franchises lies in the foundations they rely on for their story telling. It is also tied by the primary medium they serve; namely movies in the case of Star Wars, and television in Star Trek.

      Star Wars is an epic saga of warfare and adventure by a few individuals embroiled in a galaxy wide war. Battles are spread out along different isolated planets, encompassing major fleet actions and yet well within the visual range of each other, sometimes only a few 10s of km or less. Moreover,  the principal weapon used, or at least actively shown in the thick of it, is the fighter. Capital ships while visible don't seem to be as much a part of it as the fighters are, in spite of the fact that they are occasionally shown to be more than capable of taking each other out. When one looks at it, it does remind one of World War II movies, and indeed the destructive capabilities of the fleets in action do seem to reflect those seen in classic war films like Sink the Bismark, Pursuit of the Graf Spee, Flying Tigers, Flying Leathernecks, Tora Tora Tora, The Battle of Britain and  The Bridges at Toko-Ri. George Lucas made no bones about the fact (at least initially) that his stories were based on 1930s serials and WWII movies. The parallels to the first Deast Star attack to Dambusters (which includes direct transcriptions) and 633 Squadron are undeniable. So, it is not surprising that the basic mode is more like the Pacific in WWII. There are bases on planets (islands) that can be assaulted by heavy starships (battleships and cruisers, Longest Day) that can in turn be defeated by extremely heavy land based ion cannons (Guns of Navarone), other heavy starships (Sink the Bismark), and more generally by small fighters and light bombers (Tora, Tora, Tora). In the mix you throw in the MF which is like a PT boat. The destructive power is in that sense is similar in order to have similar action sequences as the ones seen in those kinds of films. We are not talking kilotons here.
     Star Trek, on the other hand, doesn't even have the word "war" in the title. As a result the "E" is generally only as powerful as necessary for a given plot. It is also not generally used as part of a fleet, so it needs to be able to defend itself independently. The 19th century sailing adventure stories where single ships generally only slugged it out with other individual ships of a similar class - armed worldwide cruisers - is the principal model for this type of story. Gene Roddenberry professed that Kirk could be seen as Horatio Hornblower in space. Fleets of this period patrolled alone in far off oceans typically  protecting trade, initiating contact with new cultures, first contacts with islanders, colonization, reconnaissance, and just plain scientific exploration. When there was warfare, it was intense, but slow, allowing the captain to get advice from his first officer on best tactics while under fire. Federation starships can withstand detonations of incredibly powerful weapons at point blank range, which like the way pirate movies have ships can fire broadsides at each other and yet some how, Error Flynn can still command a ship.
    Star Trek stories tend to start when you get somewhere and end just before you leave. For the purposes of plot, you can't get help from anywhere particularly quickly or run away quickly so the vastness of space is a plus to explain why you have to resolve your story within the confines of a 1 hour episode, basically exactly where they are. The 19th century sailing adventure fits that bill rather nicely. Unfortunately movies, and an increased demand for faster paced stories has somewhat killed that. Star Wars is really the epitome of space opera. Space opera is by definition epic, but as a movie it has a limited amount of time to tell a story. Space Opera also require action. Things have to happen, one right after the other, so you can't have "all that tedious mucking about in hyperspace" type stuff. Vast distances of a galaxy have to be covered in less time than what it takes for the next bathroom break. The threat has to be eminent, and not surprisingly armed to the teeth. Star Trek has not escaped this as the movies have also relied on much more warfare and destruction to keep the action going. The need for speed has even been epitomized by the last Star Trek movie which presented negligent travel times between Earth and the Klingon home world either by ship or nearly instantaneously by transwarp transporter (even instantaneous communications; the days of Uhura saying it will take several hours for a reply from Starfleet are gone). Whereas in the original series starships were rarely destroyed in warfare, battles where much more the norm in the DS9, Voyager, and Enterprise incarnations.
     So how powerful are they? Well in the end, really only as powerful as they need to be. To expect consistency, particularly from the film franchises, is chimeric as well. The levels of technology are as Arthur C. Clarke would have it, no different from magic as far as an observer from the late 20th or early 21st century would see it. There is definitely a certain enjoyment one can have from trying to figure out how many Gigawatts a weapons system generated or the number of g's pulled by a ship during some maneuver, but in the end I really wouldn't expect the writers of these franchises to put anywhere near the effort to figure it out and film it accurately. From their point of view, they want to tell a story in the best way possible. If that means you can hear explosions in space, so be it.