Here goes my long answer...
The comparison between Omega and Rolex is decades old, and produces emotional responses, but trying to take a more objective view, there are things to consider:
Let's see a very long answer: Both brands make quality products, no doubt, some of the best in the market. And the slightly more expensive prices in Rolex, in some models, not in all, is not due to a better quality control investment. In fact, there are famous quality control flaws in Rolex, not common, but quite impressive, like watches with two 3 numbers, one also at 9… The slight difference in prices is better explained by Rolex being more trendy and with much higher marketing.
In a comparison between Rolex and Omega, I would list some points. I will begin saying I like both brands very much, but…
1. Omega has much more history, like 60 years more. It was already a very well established brand with high reputation for innovation, quality and robustness when Rolex was beginning.
2. Omega has in-house movements since the XIXth century and has only used in-family movements for two decades, part of the 1980s and the 1990s, but never stopped producing some in-house movements. Rolex never made a single movement until very few years ago, already in the XXIst century. All of them were off the shelf Aegler's, just as others used ETA, like Breitling did for so many decades until recently. Rolex in some decades used also other brands too, like Valjoux’s and Zenith’s. So, Rolex movements made by Aegler were not even “in-family”… And that is true for all the famous movements inside Rolexes from the 1960s and 1970s, not Rolex, Aegler’s.
3. Omega has produced high horology, some of the principal high complications in horology, since the XIXth century, minute repeaters and perpetual calendars, and since 1947 Omega produces one of the most complicated challenges in horology, Tourbillons. And not just a Tourbillon, but a central Tourbillon, of which, by the way, Omega produces now not one but two different calibers, and has produced, several more along its history. Very few brands can make, know how to produce and want to produce Tourbillons. Omega is one of those. Rolex has never produced high horology. Not only that, Omega produces a solid gold Tourbillon movement. Gold movements are produced by only a few, like VC, FP Journe and Omega.
4. Omega made a Group Company, SSIH, with Lemania. Two brands of the same company and together designed and produced some of the most refined chronograph movements of horological history, like the 321, again produced by Omega today, a caliber used by Patek Philippe, Vacheron Constantin and Roger Dubuis, among others. In fact, caliber 321 is in great part responsible for the continuation and resurgence of mechanical horology since the quartz crisis. Other Omega chronograph caliber (nothing less), like the 1861 has also been used by Vacheron Constantin, and also by Franck Muller... Also, Audemars Piguet has used at least one Omega in-house caliber, time only. So, Omega calibers have been used by Patek Philippe, Vacheron Constantin, Roger Dubuis, Breguet, Audemars Piguet, Chopard, Zenith, Cartier, Franck Muller and are the base for the amazing and artisanal hand made Atelier de Chronometrie watches. The reason for this is their extremely high quality and qualities. Aegler's and Rolex movements were never used by a single one of those brands, only by Gruen. And there are good reasons for that. Rolex couldn't made their own chronograph movement (neither a simple time movement) until the 2000s, so they used Valjoux, which are very good, and Zenith's, also very good. But it is clear that horologically they didn't have the capacity Omega had for decades and even more than 100 years before Rolex began producing their own movements. Omega had in-house movements, time only and chronographs, even in the pocket watch era, in the XIXth century and chronographs before their association with Lemania. In fact, caliber 321 is strongly based on an Omega caliber previous to the association with Lemania, and was produced mostly because of an Omega initiative. Very few brands were capable of producing their own calibers and more so their own chronograph movements. Longines and Angelus did it. Of course, Omega. Not even Patek Philippe, Vacheron Constantin or Audemars Piguet could, they began making their own movements just until the Swatch Group stop selling them the Omega-Lemania 321 (to PP and VC; AP used also a Lemania module). The new chronograph calibers by PP and VC are completely based on the Omega-Lemania movement.
5. Omega is a pioneer in technology, such as the Co-Axial but also of the finest quartz movements, and of materials. The first time a company used the stainless steel Rolex uses today was Omega in the early 1970s. The Megaquartz from the early 1970s were some 15 times more accurate than the Oysterquartz from 2001…
6. It is one of the very few brands to engage in a new escapement, and to this moment one of the most different and high performance escapement technologies in centuries. Breguet has also done marvels, but only in very few numbers. Omega has achieved this for large production, not easy.
7. Rolex made their first dive watch in 1953, the Submariner, 100mts water resistant, and strongly based on the Blancpain from months earlier (that’s another interesting story). Omega made 22 years earlier a divers watch for 135mts deep, the Marine. When Omega was producing Seamaster 300s good for 200mts deep, Rolex was making Submariners 100mts WR (it took Rolex several years more, already in the 1960s, to make them 200mts WR, something Omega achieved in 1957), and so on those differences are maintained with Omega always producing more performant technologies and watches. Simply put, better watches. The most extreme divers Rolex today is a Deep Sea, good for 11,000mts depth (until some months ago it was 3,900mts depth). The most extreme divers Omega since some years now is capable of 15,000mts depth. The Rolex Milgauss, is specifically made to be non affected by magnetism. Rolex doesn't publish the actual data (their webpage states a dubious 1000 Gauss, it should be more), but being a brand that publicizes so many doubtful data, like being the first to produce a divers watch (literally said in their webpage and completely false), being the first to produce a GMT (wrong again), etc, one would expect they would publish a better result than Omega if they had it. Any ladies mechanical dress Omega stands 15,000 Gauss. But Omega doesn't do the marketing Rolex does.
8. The finishing of the movements in Omegas has always been superior to the industrial finishing of Rolex (Aegler’s) movements. Just comparing, for example, calibers 5XX, 55X, 56X or 75X to their Rolex counterparts, calibers 1066 or 1575, will quickly evidence this. But the same today with 8500 vs 3231.
9. Rolex main "inventions" have been old or not patented designs from others. The Oyster case is based on somebody else's patent and even manufactured not in Rolex installations for many years. So, not really their idea, not really their production... Full rotor automatic system is an invention of the late 1770s, but Rolex made their patent in the XXth. To me Rolex is more admirable by what they have achieved with great marketing, although the product is good, but not extraordinary as they make it come across. It is a quality watch, below in history, refinement or technical specifications to Omega, and of course below JLC, but they have made it a trend for people to think that Rolex is at PP's level (!). I prefer a brand that tells you clearly that they use ETA movements and never invented anything but they do things with quality, like for example Stowa or something like that, than a brand that sells itself as if it was at PP's level and is in reality just above Tudor and below GP and Omega. Their marketing is clearly top notch and has produced a distorted perception in a wide public. The first thing the guide at Patek Philippe’s Museum in Geneva tells you is, “Let’s get something clear, PP is a brand that competes with Vacheron Constantin or Audemars Piguet, even with JLC, but in no way Rolex competes with these brands.” True story. They have to be clear about it because there are people that do think Rolex is at such high levels, and that is just pure marketing.
10. Omega was perceived as a better brand until the quartz crisis, but Rolex managed much better the crisis because they had moved years before, in the 1960s, from a broader market brand to only luxury. They produced gold-plated watches before, but stop doing so in good time. Omega management in the 1970s, with the crisis, though the opposite, that having a broader market would be beneficial in the crisis, so they begin to make some gold-plated watches, which was very rare for them before that. In fact, those that passed the crisis best were the only premium brands, all others had to compete with much lower prices from Seiko and Citizen.
Nevertheless I find the 1970s to be an interesting moment in designs, and also in calibers for Omega. Omega's best movements are often reckoned as some of the very best movements of all the XXth century up until the end of the 1960s, but, in fact, 1970s Omega calibers, (Family 1000-1010, 1020, which were high beat and produced since 1968) were more trouble free than the first Rolex high beat, 3035, from 1977, 9 years later. They had to correct problems along the production through several years, until it was more robust. Today Omega is producing really amazing things, much more than in their highlight in the 60's, more strongly high tech, more differentiated technology from anybody else in movements and materials, and the quality hasn't been better.
11. Omega has produced several iconic designs, from Gerald Genta designs, not one but several, to the Speedmaster and some of the very few 1980s iconic designs and which survive today. Also it was Omega the first brand to produce the stainless steel integrated bracelet casual watch, which many copied the idea afterwards, and even Rolex took the design of the Oysterquartz quite literally from a 1972 Omega design.
12. Rolex has great marketing, better than anybody else's, and better brand structure than Omega. In great part because of that, also much better resale prices than most. Omega has several lines and in each one you find chronographs, or dress watches, etc. That is confusing even for those that know the brand well. And Omega's marketing is not the best. But Omega has much more technology, refinement, history, substance. In fact, the only brand to be able to produce 15,000mts WR watches, 15,000 Gauss watches, Tourbillons, annual and perpetual calendars, their own chronograph movements (not only one caliber but 3 different in-house chronograph movements, and at times four), their own innovative escapement, dress, sports, technical watches, high horology and always a very homogenous excellent quality, the indisputable best thermocompensated quartz tool watches (the one approved by NASA, ESA and Russian Space Agency), the only brand to do all that is Omega. That is my view based on technical and historical aspects.
I believe that in all of these points we can see a knockout, but Rolex produces a good quality product and an extraordinary marketing, good for them. If selling more is THE point to measure any brand, Breguet produces 20,000, Vacheron Constantin 30,000, Patek Philippe only 60,000 pieces. There are many other important aspects to consider, if we want to go deeper and not remain in a superficial analysis.
All this is in comparison with Rolex. It goes without saying why Tudor doesn't compete technologically or in refinement with Omega. Tudor competes with Longines (with which it isn't a clear winner), with Breitling and Baume et Mercier.