Let me try it from this angle:
The stick accelerates because of force exerted by your hand. That's Newton's first law of motion.
So if your hand is exerting force on the cue, then it would stand to reason your hand is still in the process of exerting force DURING contact, however brief contact may be. This force will create a tiny amount of additional ball speed to add to the momentum transferred by the cue's 20 mph x 19 oz. of momentum.
Compare this to a stick moving at constant velocity. If the stick is moving at constant velocity, your hand is not applying force to accelerate it, so it stands to reason your hand is not exerting force during contact. The cue ball speed will be generated only by the stick's momentum.
The force exerted by your hand DURING contact transfers momentum to the cue ball through IMPULSE, which is defined as force X time, which can be set equal to mass X velocity. For example: 2 newtons of force applied for one second will accelerate a 1 kg object to a speed of 2 meters per second, or numerically 2N X 1s = 1kg X 2m/s. In the case of the cue stick, the time factor is very small. If you can exert 5 newtons of force during your stroke, and the cue ball weighs .15 kg, and contact is only .005 seconds (I'm pulling those numbers out of nowhere), then by the equation 5N X .005s = .15kg X .17m/s, you're only ading .17 m/s of cue ball speed to the ball, which is a small difference, and the reason it's so small is because your 5 newtons is only being applied for .005 seconds.
I probably just used a lot more physics than necessary, but that's why an accelerating cue stick should hit the ball slightly faster vs. a "coasting" one.
-Andrew