thrust remaining more or less equal over all airspeeds for jets and tapering off for props, and power increasing with speed for jets and remaining more or less steady for props
We must be very careful in making this type of comparison.
Brake Horsepower is proportional to torque x RPM.
Propulsive power is equal to thrust x TAS.
If we use the simplifying assumption that jet thrust is almost constant at all airspeeds, then as we increase the airspeed the constant thrust multiplied by increasing TAS will produce increasing propulsive power. But the turbojet engine will continue to produce no BHP.
The BHP produced by a piston engine is almost constant at all airspeeds. But the thrust produced by the prop gradually decreases as TAS increases. So as we increase airspeed, the decreasing thrust multiplied by the increasing TAS causes the propulsive power to increase up to some maximum value then decrease as airspeed continues to increase. The propulsive power curve for a prop is a bit like an inverted drag curve.
So overall we can say that as airspeed increases:
PISTON PROPS produce
Constant BHP.
Decreasing thrust.
Propulsive power which increase up to some maximum value then decreases again.
TURBOJETS produce
Almost constant thrust.
No BHP.
Increasing propulsive power.