The escapement in a mechanical watch is affcted by what is known as "positional error", which is basically the effect of gravity on the escapament mechanism. What this means is that a mechanical watch will run at slightly difefrent rates depending on the position (attitude) of the escapment, (i.e dial facing up, vs dial facing downm, or crwon facing up, vs crown facing left). A watch can be adjusted to correct for some positional error (usually done to 4 positions, dial up, dial cown, crown up, crown down), but due to the many angles a watch goes through on the wirst, its almost impossible to completely counter positional error. A tourbillon of the ordinary type (rotates in one plane only) is designed to help counter positional beat error. They GyroTourbillon is an extension of that premis, in that it rotates through two axes of motion. Theoretically, this should compensate for any positional error.
Having said that, how much is positional error? It can be as much as 6-10 seconds per day when measured between positions (ie statically measured, not as worn). In a really well adjusted watch these errors average out, so that the over all rate per day may be very good. Indeed, the COSC Chronometer spec calls for a rate between -4 and +6 seconds overall.
Most COSC certified watches (eg Rolex, Omega, etc) run in the one to four seconds error range per day. The Gyro Tourbillon, from what I have read, cuts positional errors donw to under 2 secinds per day regardless of position. That should make a more accurate watch overall..
That's the theory, but also remember that the high end watch companies also try to out-do each other with these high end features. I know that the next step after the original Gyro Tourbillon, was to and an equation of time complication to it (that shows the difference between the "Sun" time, and "clock" time..) All this adds up to great marketing, and in most cases, some of the advances worth their way down into the consumer models. Case in point, Omega's coaxial escapement movement, started off as an exclusive item, but now is pretty much the way of all Omega movements..
Hope that helps, and you have to admit, its a pretty stunning piece of horological engineering..