Really just parroting whats already been said...
Both are great airlines.. Ive flown more on Emirates over the past 15+ years than I have Qatar.. but over the last 3 years Qatar has definitely been my preference..
Qatar makes firearms a bit of a PIA.. but if you have a travel agent, they will handle that for you.. and if you don't, there are plenty of people here on AH that can help you through the process (I've managed it twice at this point.. its really not a big deal.. just a pain at times)...
Out of DFW both airlines are going to have a significant number of Indians, Pakistanis, and middle easterners on the flights in coach.. Not really a big deal.. but by the time you land in Doha or Dubai the toilets will be a disaster zone, the aisles will be filled with trash, etc.. The trip we made on Qatar to Joberg back in July, it was frankly disgraceful how bad the plane looked by the time we touched down in Doha.. but honestly, who cares? I felt sorry for the guy that had to clean the bird after we got off.. It looked like 200 pre-pubescent teens had been watching the Rocky Horror Picture Show for 14 hours straight in the economy section.. but I spent the entire flight either sleeping or watching a couple of movies.. so it had zero impact on me personally (other than making sure I put my shoes back on before going to the toilet.. I had no desire to stand in 25 peoples piss that was all over the floors)...
To their credit, seats are more comfortable and more spacious than American and United options (both Qatar and Emirates)..
For me, ALL air travel sucks these days.. the concept of customer service seems to have left the industry entirely.. prices have gotten crazy (as with most things on the planet it would seem).. etc
But.. For me.. when traveling internationally, I almost exclusively avoid US carriers whenever I can (I say that.. but I'll be on American to Dublin in a month.. and then on American to Buenos Aires in 2 months)... If I am flying west I try to find flights on Cathay Pacific, ANA, Korean, or just about anyone else I can find other than American, Delta, or United.. When flying to Europe I do my best to catch Lufty, Air France, BA (although I think they all suck as well).. and when flying to Africa I do my best to go Qatar as a priority, Emirates is my second choice, and then literally will take the Ethiopian or Kenyan option before I fly with a US carrier anymore..
Thats a pretty sad statement when I'd rather fly Kenya Airways than I would United.. But.. it is unfortunately true (I cant tell you how many times I have flown from DC to Nairobi on Kenya.. and then caught a regional flight to get to my final destination... rather than having to get on a United/Delta/American bird...