Here is an introduction:
Airborne Wind Energy - Producing Energy with Kites
Sky Windpower style lifts the entire wind turbine into the air, which is heavy, and can’t sweep as much area, so can’t produce as much power, it’s not even close IIRC. I think that’s the main reason it is not pursued more, as the goal is to produce electricity. Also the electric cable to the ground becomes heavy. Lifting heavy things needs a strong wind so your capacity factor, what percentage of the time you’re generating electricity, becomes lower. There’s also a lot that can be said about outdated assumptions and realities of flying in the jet stream.
Edited to add (ETA).
[ETA: Moving] Crosswind also has disadvantages, flight control is difficult, a long fast moving tether introduces a lot of drag, and with ground generation maximum power is intermittent (about a quarter of the time IIRC) and the tether wears out.
If you [ETA: move] crosswind but put generators on the wing, you now do have continuous power but again your cable becomes heavy.
Everyone has biases about what they like best. I like multiple kites flying crosswind connected to a single tether best, but then flight control presumably becomes even more difficult.
Here are some videos of the Sky Windpower system in action: