Tudor vs Omega: Build Quality. Which is better?

I suspect there might be some "endowment effect" bias going on.  Nonetheless, I've had the exact same reaction as you, when comparing Tudor to Omega - heck, even comparing Rolex to Omega!

• I owned the Oyster Perpetual 41 in silver.  It was a beautiful and very rugged and utilitarian watch
• Nonetheless, every Omega I've owned has always felt much more "luxurious" than that Rolex, and any Tudors I ever tried on
• But, I suspect this is how Rolex and Tudor have purposefully positioned themselves in the market.  The 2 brands are well-known and well-loved for the robustness of their watches and rugged feel, yeah?  They don't aim to be luxurious - they want the tool watch aesthetic
• Meanwhile, Omega puts time and effort into frilly stuff like lyre lugs and all that jazz

So, I can totally see how Rolex fans might be like, "Why waste all that time and energy on frivolous stuff like lyre lugs?" when they try on an Omega, yeah?

I have several Omegas and only 1 Tudor (BB58). I would have to say that the Omegas tend to have a higher quality "feel" compared to the Tudor, but that is also reflected in higher MSRP.  The BB58 is an awesome watch, but it has more of a "tool watch" feel, while my Omegas tend to have more of a "luxury watch" feel.  Much of that is attributable to case shape and finishing, as the slab sided case of the BB tends to make it look kinda boring and utilitarian. I would tend to agree with @HotWatchChick69 that Omega may have a more luxurious appearance than Rolex as well, but the buttery smooth bezel and crown action on Rolex is superior to Omega in my opinion. 

You will find much debate on this. I have multiples of each and find them to each have their strengths and opportunities. My observations of each.

• Tudor - More experimental in their design language. Great finishing, solid movements with top tier accuracy, with comfortable bracelets. Strong tactile feel for both bezel and crown. Cons, the looks are not for everyone and as a result of their experimentation they have many misses.
• Omega - Classic looks, also excellent movements with better finishing. Accuracy and metallurgy seems and feels the same. Bezels are absolutely meh, but the crown action equal.  Cons? The bracelets should be much better designed for the price. I agree that they feel more luxurious, but not by much.

Both are excellent choices regardless, with Omega getting a slight nod in my book. Here is my observation of the market and not of the watches themselves, so take it with a grain of salt; Omega is trying to play catch up with Rolex, but Rolex lets Tudor to compete with Omega by undercutting comparable Omegas on price instead of being bothered themselves and by market growth, they seem to be succeeding.

#omega  - Quality, functionality, variety brand recognition, NASA approved (speedmaster) and James Bond approved (Seamaster), Always available at AD. 
 

#tudor  - Rolex’s sister company 

I have both and love them like my kids. Omega is my favorite 

There is only one answer, Omega.

What constitutes build quality to you? Robustness? Finishing? Tudor and Omega seem to represent opposite ends of that spectrum.

I think Omega is better and should be better as they are prices roughly 1.5x to double what the Tudor comparisons are. 
I have an SMP And felt exactly as you did when I tried on the BB58. However I tried on a Ranger recently and that felt much better than the BB58. Maybe I just realigned my expectations since. 
I’m still waiting for the Pelagos 39 and if I’m not happy with that I will get myself the ranger. 
 

Omega is the answer, as it should be for the price difference. Just throwing this out there, but if you’re in the market for a BB58 but also feel underwhelmed by it in person, your gut is trying to tell your wallet something. Listen to your gut and not the hype.

I have a Ranger and I love it, but the BB58 felt more or less the same in terms of quality (sans the improved bracelet). I certainly wouldn't mind getting a BB58 or a Pelagos 39 down the road, although their prices are getting pretty close to Omega territory. (a new Railmaster is something like 5k EUR)

That’s true. However in the Uk omega just just jumped up the prices massively this month. A Speedy went from £6800 to £7400. SMP from £4800 to £5200 for example.

So I think the difference in pricing will still be there.

Since my comment last, I actually turned down a Pelagos 39 in favour of the BBPro. This is by far my favourite Tudor watch now.