The carriers were supposed to get the new American electromagnetic catapult and arrestor system. However, that has turned out to be a complete fiasco so far, so it seems like Britain dodged a bullet on that one. The Americans will probably sort things out eventually, but the problems would have reduced Britain's new carriers to helicopter carriers only in the mean time.
However that was luck. The decision actually came down to someone in the MoD doing the sums and finding out that when you factor training and the rotation of pilots into the equation, the 'B' (vertical take off) version was much cheaper for Britain to operate.
"Conventional" take-off and landing on a carrier takes constant practice for the pilots to retain qualifications, and it ties up a carrier while they do so. All of that cost loads of money in fuel. salaries, and equipment hours. Britain plans to operate the planes and pilots from a common "pool" with the RAF, to provide more flexibility and to ensure the carriers aren't dependent upon a very small pool of dedicated naval pilots.
The short "rolling" take off and landing (they won't actually use vertical take off or landing) capability in the F-35B is nearly automated, and is simpler than it was with a Harrier, and vastly simpler and easier to learn than catapult and arrestor hook equivalents. The pilots can rehearse this on land airfields (equipped with a ramp for this purpose), which means that the carrier can be on active operations rather than tied up in training maintaining pilot currency. The carriers can operate with 12 F-35Bs under normal circumstances, but "surge" to several dozen more as circumstances require, and all without having to maintain a dedicated pool of specialised naval pilots.
So overall, given the UK's particular situation, they decided to go with the solution that saved significant amounts of money, provided more operational flexibility, and didn't tie up a carrier as much with training. But the money saving was the big one.
P.S. - When reading about costs in the press, keep in mind that the MoD does full life cycle accounting these days, which includes fuel and salaries, which together can greatly exceed the sticker price of a plane. You have to dig to find out what those are however, as the popular press often don't understand what those are and just publishes a "big number" and let's you assume that is the sticker price.