Thanks Mana for starting for a great STS thread, but also Vessbot and FCeng84 for some very high quality explanations of speed stability and augmentation.
I don't have much to add especially to Vessbot's original Post#5, but I do have an interest in a general question: can a feedback system based on speed input and stab trim output like the STS actually create a longitudinally stable aircraft, or is it just simulating one for certification reasons? I would argue the answer is somewhere between the two.
Creating a longitudinally stable aircraft (in oversimplified terms) is achieved by placing center of gravity in front of a calculated neutral point. Highly stable aircraft effectively waste lift and lower efficiency by generally having their horizontal stabilizer in negative AoA/lift, which is fine on a C172 at forward COG limit but wasteful on a modern air transport. Maximum efficiency should be achieved at cruise if aircraft COG is assigned so tail AoA is approximately 0'. At cruise a 737 will be operating at a few degrees of wing AoA so the aircraft will still be stable, but probably marginally so, as compared to certification requirements originally developed for an older generation of aircraft. The closer the aircraft gets to neutral stability, the more problematic it's control characteristics become. At true neutrality the AoA of the tail and wing are equal, and the aircraft is effectively trimmed simultaneously for any AoA/speed (i.e. it is just as suited to VNE at a very low AoA, or stalling AoA at low speed). Not to mention that the tail might stall before the wing.
FAR states that a transport airplane should exhibit positive speed stability with a stick force of at least 3 lbs per 10 knots, one characteristic of a longitudinal stability that the STS assists with achieving. As others have noted, as thrust in the 737 is increased the plane will tend to not accelerate but instead pitch up and settle to the same speed in a climb (thrust line effects are a factor here also) just like a C172. As other posters noted it just feels odd that if you want to accelerate in level flight you will have to remove some trim that the STS just obviously applied against your intention.
However, is the aircraft now fully longitudinally stable in the same way the C172 is? The COG has not changed position, and the AoA differential between tail and wing is still low (i.e the same just-stable configuration as before). The aircraft is not any more resistant to pitch changes from updrafts and downdrafts. The C172 has more pitch stability in this sense as the greater difference between trimmed wing and tail AoA means the tail creates a stronger opposition to an airflow disturbance that increments AoA equally at the wing and tail.
The fact the STS has met the speed stability requirement for certification, but only created effectively half of a stable aircraft, is more of an interesting theoretical issue than a real one. The aircraft will be likely to have sufficient pitch stability to make manual flying perfectly acceptable and pleasant. However, I think this would change if pilots had to fly whole sectors in STS assisted manual mode, hour after hour and through turbulence. Like a 1940s transport pilot, there would be a strong push to demand aircraft loaded further forward with stronger pure aerodynamic longitudinal stability.