SPIN
Gyroscopes generate what is called gyroscopic rigidity. This is basically the force that prevents them from leaving their current plane (and is what allows a disc to fly without tumbling out of control). In the case of a disc it fights pitching and banking. The higher the spin, the more gyroscopic rigidity a disc will have. I.E. The faster the disc is spinning the more it will fight against both turn and fade, or another way, the faster it is spinning, the less it will turn OR fade.
There is a common misconception that lots of spin will increase high speed turn, because the left side of the disc is flying through the air faster than right side, thus generating more lift on the left side than the right, thereby imparting a "bank to the right" or "turn" torque. This is completely wrong on a number of accounts.
#1 The left side of the disc and the right side of the disc are flying forward through air at the same speed (unlike a helicopter where you have 2 separate wings, one flying forward and one flying backwards). The lift generated by a wing is caused by the displacement and deflection of air. The leading and trailing edge of the left side of the disc are passing through the air at the same speed, at the same angle, and displacing and deflecting air in the same fashion. This is what creates the pressure differential between the top and bottom, which generates the lift.
#2 Gyroscopic precession. This one always gets me because people oftentimes correctly describe gyroscopic precession when explaining fade, but then totally forget about it when trying to explain high speed turn. Gyroscopic precession basically alters any torque 90 degrees to the right (for RHBH), so a pitch up torque is manifested as a fade to the left, pitch down, turn to the right, and so on. If in fact the left side of the wing was generating more lift than the right side, this would result in a bank to the right torque, which due to gyroscopic precession would cause the disc to pitch or nose up. Observationally this is not the case, because in fact the disc turns to the right. In order to generate a turn to the right, there needs to be a pitch down torque applied to the disc.
#3 The only paper I ever read that tested a disc in a wind tunnel (Potts and Hummel) did not observe any measurable pitching or banking due to spin.
VELOCITY
A disc released with more velocity will always have more initial high speed turn than the same disc released with all the same release characteristics but less velocity. It will also reach it's fade point later in flight.
This is because of what causes fade and turn. Every disc has what I will call a neutral angle of attack, where the center of pressure is directly in the middle of the disc (exactly where the center of gravity is). At this angle the disc is being pulled upward and downward from the same point so there is no torque forces being applied. At a higher angle of attack the COP is forward of the COG which results in a pitch up torque, and at lower angles of attack the COP is behind the COG resulting in a pitch down torque. Due to gyroscopic precession, the pitch up torque is translated into a bank left or fade, and the pitch down torque to a bank right or turn. (Interesting tidbit: This is how helicopters steer. There is a linkage between the stick the pilot uses that alters his inputs 90 degrees. So when the pilot pushes the stick forward, the linkage causes the right side to generate more lift, which due to gyroscopic precession results in the helicopter nosing or pitching down.)
Lift is directly related to velocity, more velocity equals more lift. So let's look at 3 examples of the exact same throw but with different release velocities: an amateur throwing a disc, your grandma throwing a disc, and a pro throwing a disc. We'll assume the neutral angle of attack for this particular disc is 10 degrees and all 3 are releasing it at a 10 degree angle of attack. So at initial release for all 3 of our throwers, there is no torque forces being applied since the COP and COG are in the same spot.
Let's look at the Am first. We'll assume the am throws the disc at a velocity where the disc is generating the exact amount of lift to counteract gravity. So the Am's disc flies out a short distance and has neither descended or ascended, so it's maintaining the initial 10 degree angle of attack. At this point the Am's disc is still going straight, and has neither turned or faded because it is still flying at the neutral angle of attack.
Let's look at the Pro. The pro has thrown the disc at a much higher velocity, such that the force of lift exceeds the force of gravity. This results in the pro's disc ascending. So the pro started with a 10 degree angle of attack, and the ascent has caused it to start traveling forward and up at an angle of 5 degrees. So now the pro's angle of attack (keep in mind the angle of attack is the angle through the air, not in relation to the ground) is now only 5 degrees. Due to the decrease in angle of attack, the COP on his disc has moved behind the COG, resulting in a pitch down torque. Due to gyroscopic precession this gets translated into a turn to the right.
Now let's look at Grandma. Grandma can't throw hard, so her disc is generating less lift than gravity resulting in her disc immediately descending. We'll assume shortly after release it's descending at 5 degrees. Couple this with the initial angle of attack of 10 degrees, and now grandma's disc is flying at a 15 degree angle of attack. Due to this increase in angle of attack the COP has moved forward of the COG, resulting in a pitch up torque. Due to gyroscopic precession this is translated into fade left.
Of course due to drag, everyone's disc will slow down and gradually lose lift. Eventually the pro's disc will start to descend and begin to fade back left, resulting in the beautiful long distance S curve. Unfortunately for grandma, her's will just fade off and crash into the ground.
TO SUM THINGS UP
More spin = more rigidity (I.E. it will resist both turn and fade)
More velocity = more initial high speed turn, and fade starting later in flight
FURTHER RAMBLINGS
All of this velocity and spin physics is why as an amateur you should try to use disc speeds that fit your power. Higher speed discs are designed to go faster. In order to do this they design the disc to be much more knife like to reduce drag, but this also results in less lift. It's similar to the difference between an airplane wing with its flaps up or down. With a high speed disc you need to be able to achieve a much higher release velocity in order to achieve enough lift for it fly a proper line.
