Performance-wise (to the ear), thinner rails tend to have more complexity/edge/buzz and thicker rails tend to be more fundamental and warm - even dull if the rails are too thick.
Playability-wise, thinner rails are LESS forgiving if there are issues with the player's embouchure/airstream, or if there are issue with the mouthpiece facing or general execution. Also, thicker rails are more tolerant of a reed to tip rail mismatch.
Additionally, the tip rail execution is more important than the siderail execution. While it's nice to see the left and right siderails with matching widths (visually), it really doesn't mean squat to the sound. With siderails, the curve is what matters the most.
So basically - the thinner the rail, the more precise the execution of the rest of the piece must be.
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