This answer relates to "Traditional Airlines" like BA, not the Low Cost carriers like Ryanair.
Forget the web just for a moment. Each flight has 1 or more different class cabins (e.g. First, Business, Premium Economy, Economy). Each of those classes has a number of booking codes represented by a letter of the alphabet. The prices are associated with those letters. The airline can assign any number of seats to a given letter on a specific flight/day. So one day letter A might cost 100 and the next day 90 and the next day 150 and the next day they may decide not to sell any seats at letter A fares.
When a person in a call centre makes an enquiry he will be shown the number of seats available at that instant. If someone else asks for the same flight at another screen 1 second later the response will be identical and it is quite possible for both users to see there is just 1 seat left. Computers love order so if both those users try and grab that 1 seat at the same time the computer will decide which one comes first and one person will get a seat and the other won't. Mind you if the user who has grabbed the seat doesn't complete the transaction (the customer says no thanks) that one seat goes back into the A booking code for someone else to book.
Now let's look at the web. Exactly the same principle applies - two users can see the same response but if they both try and book the one remaining seat one will get it and the other won't. Except it's more complex than that.
The problem is with the increase in the number of transactions that the web generates. The airline mainframes may be fast and have lots of power but their ability is finte. To avoid being overwhelmed the airlines (and the GDS) have contracts with users that impose what's known as a "look to book" ratio. In other words everytime the website makes an enquiry to the airline system it is counted. Every booking is also counted. Divide the number of bookings into the number of enquiries and you get a ratio - the websites are given a number that their ration must not exceed or there will be a bill to the agency for the overage.
So what happens is that the websites cache. They store information about each query they do make and when a query shows the flight is well and truly open they store that fact and when someone else makes an enquiry for the same flight/date/class they look at that stored data (in the cache) and use it rather than sending a new request to the airline. The cache has a lifetime - you can't assume that a request you made three months ago is still valid but the fact remains that when you get an offer it may have been sourced from the cache. The inevitable result is that however clever the cache is you may get an offer and when you say "yes please" and the website goes to the airline system the airline may say no.
Now lob in the fact that the airlines constantly review their pricing. I can't remember the precise statistic, you'll find it somewhere on the ATPCO website, but the number of fares filed and updated every single day is in the multi millions. Those fares are loaded by the airlines to ATPCO, ATPCO redistributes them to other airlines and the GDS those bodies then process the data from ATPCO so that it firs their systems and finally it appears on the airline systems. But some websites have special deals; some of those go via ATPCO but others go direct. It's is totally possible for one website to have a deal for a given seat and another website to have a different deal for the same seat and a 3rd site to have no deal at all and never even see that booking letter. And for those deals that the airlines send to websites sirect the speed to market depends on what systems the website is using and how quickly they update their databases.
Do the travel booking websites use cookies? They might but I doubt it. The complexities and dynamics of the system are such that you don't need cookies to get the sort of results that make people believe cookies are involved.
Now.......
IATA has recently come up with something the call "New Distribution Capability". I'll leave you to read and make your own mind up about that.
IATA - New Distribution Capability (NDC)