As far as I can tell, both universes have equally objective ideas of speed. Do we know how fast the two ships can go? Which is faster?
The Millennium Falcon is (approximately) 58,000 times faster than the USS Enterprise-D.
The (Galaxy Class) Enterprise-D's top speed is stated to be Warp 9.8.
DATA: Projection, sir. We may be able to match the hostile's nine point eight, sir. But at extreme risk.
This handy reckoner from the TNG Technical Manual shows us that this is equivalent to approximately 3000(ish) times the speed of light.
By comparison, the (YT-1300 light freighter) Millennium Falcon traverses a distance of approximately 70,000 light years (based on the distance from Tatooine to Alderaan, the known size of the Skyriver galaxy and the known length of a year in the Star Wars universe) in less than a few hours, equating to a top speed of around 20,000 light years per hour or something like 175,200,000 times the speed of light.
They travel at the same speed: 1.0 SOP, which is not comparable
All star ships in all science fiction are always traveling at The Speed Of Plot.
This may sound snarky but the comment is serious; it is meant to point out that the numbers are irrelevant and usually not comparable, even within the same universe.
Transportation in a story is either an unimportant inconvenience that just needs to be dealt with — simply because the audience will not accept that characters just magically pop from one place to the other (this is why Transporters were added to Star Trek) — or it is a part of the plot, by which it means that the travel from one place to the other will take just as long as is needed to make the plot come out right.
Think about it, and ask yourself this rhetorical question: when was the last time that travel in fiction happened so fast that the characters were left with lots of dead time, or so slow that a plot-line was obstructed by it? Travel is mentioned only by necessity and its speed only ever becomes important if it is relevant for the plot.
Another way of putting it is to paraphrase Gandalf's Principle of Wizardly Timeliness: a star ship is never late, nor is it early, it arrives precisely when the author means it to.
...which in turn means that if you start digging into this — as people have done starting with the very cryptic statement that The Millennium Falcon did the Kessel Run in a number of parsec, which is a measure of distance and not time — you will find that the speeds are wildly inconsistent and do not actually make any sense. As Valorum points out in the comments: star ship speeds are easy prey for Early Installment Weirdness. And once the hand-waving starts to try to make this come out right — "wormholes", "folding space" and what-not — then all bets are off and you cannot make any kind of comparison. The numbers may be incomparable even within the same universe.
The difference between "getting from A to B", and "to travel at a speed"
Note that there is a huge difference between being able to get from one point to another in a set period of time, and to actually propel yourself at a certain speed.
You can get from one town to another at an average speed of, say, 200 kilometers per hour if you go by train. But if you were to propel yourself on the other hand, with no machinery or vehicle to help you, then the maximum sustainable speed you could attain would be about 20 kilometers per hour at the very most.
In the case of the Millennium Falcon this certainly seems true in that — first — Solo claims the ship makes it "point-five past light-speed", i.e. not more than 1.5 c, which is nothing compared to any of incarnations of the Enterprise we know, from NX-01 and on-wards. But — then — the Falcon manages to get half-way across that "galaxy far far away" in a really short matter of time. So it appears the Falcon is taking shortcuts or piggybacking on something that allows it to get to the destination in a time that makes for a higher average speed than the ship can actually propel itself.
If the U.S.S Voyager could travel in the same manner as the Millennium Falcon, then the whole of the Star Trek: Voyager series would have been quite short, with the pilot episode ending with them travelling from the Delta Quadrant to the Alpha ditto with ease and being home in time for Janeway's next coffee, and that would have been it. Instead the ship plods along at 1.0 SOP until the finale, whereupon they just wormhole the ship home and close down the show.
Whereas if the Millennium Falcon would have had to suffer the constraints of the U.S.S Voyager there would not even have been a galaxy-wide war because years would be spent in transit just going from one star to the next one. But instead ships even as large as the Death Stars (120 and 900 km respectively) can just gallivant around the galaxy without even breaking a sweat.
One example that the actual values of SOP in the Star Wars universe are approaching Ludicrous Speed: the Empire has time to blow up a planet, make a dash to Dantooine, search the planet, find a rebel base, examine it close enough to conclude that "it has been deserted for some time", and then report back to the Death Star... all squeezed in between the time Obi-Wan "fear[s] something terrible has happened" and the Falcon arriving in the rubble-field that is formerly known as Alderaan.
So — again — we find that star ships travel at The Speed Of Plot. And comparisons are pretty much useless.
...with the one exception being...
The only time the actual speed of a star ship in fiction is relevant is if the plot is not fixed, but is instead open-ended. A typical case for this would be in a role-playing game.
Now I myself have not played any of the Star Trek or Star Wars role-playing games, and I doubt you will find any that are 100% canon. But if you are looking for some hard numbers regarding the speeds of the ships in question, I would direct you to look there in the first place.
This is a difficult question because if it's not a Gorilla vs Shark question, I'll have to choose sides: Star Wars or Star Trek. Where Star Trek used to give me Think Beyond feelings with "Space: The Final Frontier" and "Where no man has gone before" dialogues, I remember myself trying to behead my parents using a toy lightsaber for fun. Star Wars is barbaric and it's great that such technologies don't exist in real. Choice made. Let the battle commence!
While the other answer says that Millennium Falcon was 58000 times faster than a premium USS Enterprise, I don't agree that the Falcon won. Let me first quote a non-canonical commentary about USS Enterprise J, then I'll go with canon:
I guess I’d have to call it a Universe class vessel. The approved J had one deflector, recognizably descended from the NX. I imagine they are beyond Transwarp. I imagine they can fold space, and that they are exploring other Galaxy’s (extremely risky business) besides the Milky Way.
- Doug Drexler
(Thanks to @Valorum for it)
Now, the canonical race:
Rules: Millennium Falcon and USS Enterprise E are floating side by side at one end of the galaxy. They'll start running on the firing of Death Star hyperlaser. Whichever would hit the red ribbon on the other end of the galaxy would win. TV cameramen are on Infinite Improbability Drive to record everything.
Picard: Let's see what this Galaxy-class starship can do. Maximum Warp. Engage.
Falcon's computer is still computing before the hyperspace jump and the Enterprise is already several light years away.
Data: Our sensors indicate that Millennium Falcon just moved past us and it's 100 light years ahead of us.
Picard: Never Surrender. Never Give up.
After reaching the other end of the galaxy, Riker's (great great great)^100 grandson: Data, let's jump back in time.
USS Enterprise E hits the red ribbon moments before Millennium Falcon.
Note: I didn't cheat. If you understand Spacetime well, going FTL actually mean traveling back in time. Also, USS Enterprise E has time traveling capability without using Slingshot Effect. It was shown in Star Trek: The First Contact (Enterprise returned back to future on its own).
We can't say for sure.
It may seem like it can have an objective answer because both Star Wars and Star Trek universes show top speeds of ships. The main problem actually is that we don't know for sure if standards and units used in both universes are same.
For example, the accepted answer says that Millennium Falcon is 58000 times faster than USS Enterprise D based on the fact that the Falcon travelled 70000 light years within hours. First, do we know that light year definition is same in both universes? After all, Star Wars has a track record of using parsec as a unit of time. Second, if the definition is really the distance travelled by light in one year, what if a standard year isn't 365 days long? There are other troubles like definition of fundamental units of time itself.
There are other issues, too. We don't know for sure if physics and fundamental constants of both universes are same. And, I've a lead here. In Star Wars universe, there's this thing called "The Force" exists. What if midichlorians interact with photons to reduce speed of light in vacuum significantly?