The jet aircrafts speed stability is much poorer than that of a propeller driven aircraft because of the following reason;
Thrust changes with speed, which helps to improve the speed stability on propeller-driven aircraft and does not improve it on jet aircraft
To understand the above comments in the original post we need to look at the thrust curves for jets and props.
For conventional turbojets (with no bypass or low bypass) the thrust is a maximum when airspeed is zero, then reduces as speed increases from zero up to about 250 knots. Thrust then increases again due to ram effect as speed continues to increase. If you think about it this thrust curve looks pretty much like a standard drag curve (allbeit with less pronounced slopes).
This means that throughout much of the speed range, as aircraft speed changes the thrust changes in the same sense as the drag. So if Vmd is in the region of 250 knots then below Vmd if speed increases, the decreasing drag is matched to some extent by the reducing thrust. And if speed increases above Vmd the increasing drag is matched to some extent by the increasing thrust. The overall effect is a reduction in speed stability.
For a propeller aircraft the thrust is maximum when airspeed is zero, then thrust gradually decreases with increasing airspeed. The airspeed never gets high enough for ram effect to be a major factor.
This means that as airspeed increases above Vmd, the drag increases while the thrust decreases. This makes the aircraft more speed stable above Vmd.