The push-pull knob-on-a-stick controls originally were
vernier controls. That is, they could be very finely adjusted by twisting the knob, which screwed them in or out in a threaded mount. Pushing the lock button disengaged the shaft from the threading, allowing them to be moved more quickly by straight in-and-out motion.
McFarlane Aviation Products - Vernier-Assist Mixture and Throttle Controls
Although in my own flight training at that level, I never knew about the vernier function, just the pulling and pushing. Small planes usually don't require tweaking the throttle in 1% increments.
BTW - Cessna uses vernier knobs on the 172 and 152 airplanes.
http://www.aerotrek.aero/photos/tech/throttle.jpg
The name "Vernier" comes from the scale invented by Pierre Vernier for extra-precise measurements and adjustments:
Vernier scale - Wikipedia, the free encyclopedia
The quadrant control (lever pivoting on a fixed point) is virtually always used on multi-engine aircraft, because it is much easier to grab 2-4 quadrant levers side by side, and move them together, than 2-4 vernier-type knobs sticking out from the panel.
In addition, it is easier to put a scale of settings beside a quadrant lever or levers (e.g. Reverse, Idle, Beta, Climb, TOGA) than it is to label a vernier control.
As the linked thread from Zio mentions, larger (i.e. multi-engine) planes will always have quadrants, so some manufacturers use them in their single-engine trainers as well, to get new pilots used to the design from the beginning.
Quadrant (lever) throttles are also generally used when the flight controls are via stick rather than yoke (everything from fighter planes to the Piper Cub), since the pilots of those planes use opposite hands for controls (right hand on stick, left hand on throttle) and the throttle is mounted on the side of the cockpit. And also in tandem-seating planes (again the Cub) if front and back seats both have throttle levers, each attached to the same rod or cable leading to the engine.
http://www.dfrc.nasa.gov/gallery/pho...EC87-146-5.jpg
http://i443.photobucket.com/albums/q...hotos/j3-2.jpg
Generally, vernier thottle knobs will be cheaper and simpler, while quadrant controls are more complex (requiring the quadrant itself and the pivot) - but there are lots of non-technical reasons to use one or the other as well: habit, or perceived stylishness or "professionalism" (like T-tails on tiny trainers
) etc.