Second update in response to second edit:
The second screenshot you have now included still shows different flights. This alone is "enough said", but in case elaboration is needed:
- The two examples feature two different routes. One departs from Newark, NY, and the other departs from JFK, NY. The first one has an extra stopover in Chicago (ORD), while the second one doesn't. The domestic US flight is not even operated by Qatar like the rest of the flights, it's operated by American Airlines. I'm sure you understand why all of this can lead to a price difference.
- The dates on the two sites are different. I'm sure you know that airline ticket prices are different depending on the date.
- The travel time is different. Even if you ignored point #1 and #2, a different travel time can still make for a different price.
Putting all these points together, you see that you're talking
about two totally different products.
It's like asking why a Mercedes sold in one shop has a different price from an Audi sold in another shop.
Same exact flight, different site
Now, the following is based on my initial assumption that you were asking "Why would the exact same flight with the exact same airline have a different price at two different sites?" If this was not your intention after all, let me know, and I will remove this from the answer.
The first thing I can think of is that sites you are using to search for flights are often getting a cut or profit from the transaction. This profit can vary from site to site. As @fkraiem hinted at in a comment, this is the same with almost any trade.
Some sites forward you the airline's (or even a fourth party site) to finish the transaction, whereas others let you pay and finish on their own site. Needless to say, this leads to even more complicated factors determining the rate.
Some sites may also have special deals with airlines, affecting prices.
The air ticket market is alive, almost like a stock market. Airlines are constantly playing a game, trying to find a balance between selling a lot of tickets and getting a good price. The prices set by the airlines are ruled by hugely complex, automated computer programs. Human personnel constantly watch the process, adjusting the algorithms and do occasionally intervene manually. These processes involve monitoring and responding to the rates of competitors. Thus, if they see that a competitor has a special offer going on at a certain site, they may (whether manually, or according to an automated program) decide to also drop their prices on that site (only), in order to try and compete. This is just an example. The air ticket market is like a highly competitive game, which has gotten a lot more complex in the Internet age.
Another thing is that when you're searching for prices, what you see isn't necessarily what you get. Some sites don't immediately factor in all the extra fees that you will have to pay. Some charge quite high "credit card" fees. Many people do use credit cards, since they often get travel insurance included that way. Some force the airline to include certain things in the price. Other don't. There are a lot of sites out there. Sometimes a site may seem to have given you a cheaper price, but when you come to the actual booking, you find that your price is higher than initially predicted. On another site, the initial listing may have been closer to the truth.