The flying tourbillon is not a more sophisticated tourbillon than the classical one: it's a precise design choice with real consequences on weight, readability, and distribution of mechanical stresses. The difference from the traditional tourbillon lies entirely in the upper bridge: in the classical tourbillon the cage is held by two anchor points, top and bottom; in the flying tourbillon the upper bridge is eliminated, and the cage rotates suspended on one side only, cantilevered.
Technical paternity belongs to Alfred Helwig, teacher at the Glashütte School of Watchmaking, who patented the solution in 1920. Sixty-nine years later, Blancpain claimed the first flying tourbillon in series production in modern watchmaking. I've seen both genealogies cited in trade publications, and they don't contradict each other: Helwig opened the way, Blancpain industrialized it.
On the mechanical side, removing the upper bridge has two concrete effects. First, it reduces the mass of the cage, with benefits on the torque needed to make it rotate. Second, it eliminates a friction point. The compromise is structural: with a single anchor point, the cage is more vulnerable to shocks, and the construction of the lower plate must compensate for the rigidity that the upper bridge guaranteed. This is why in ultra-thin movements, where the flying tourbillon is almost mandatory for thickness reasons, the engineering of the lower support is critical. The Bianchet Rotondo UltraFino at 3.85 mm thick is an extreme example.
From the dial side, the circular opening that is obtained is almost always used as a functioning indicator as well: the rotation of the cage, usually one revolution per minute, is visible directly. Some calibers work at three minutes per revolution, others, like the Moritz Grossmann mentioned, even more slowly. The frequency of the balance wheel inside the cage is independent of this: it can be 3 Hz, 4 Hz or otherwise.
There are variants that further complicate the geometry. The orbital tourbillon in the Jaeger-LeCoultre 948 caliber doesn't stay fixed in the window but travels the entire dial in twenty-four hours. Inclined tourbillons, like in the David Candaux DC7, add an angled axis to compensate for gravity in multiple positions of wear, not just vertically. In both cases the foundation remains Helwig's: a single point of support, the cage free to rotate.
A note I'd like to make from the bench: many clients mistake the flying tourbillon for a time complication. It isn't. It regulates the rhythm of the movement, it doesn't add information to the dial. Its presence doesn't automatically improve accuracy compared to a well-regulated traditional tourbillon. What changes is the construction, and, if you're close to the watch, the ability to watch the cage rotate without anything visually crossing through it.