Does the power behind the engine make any difference to how strong the thrust is?
You ask: 


  If the engine can spin the propeller fast enough, why does it need power behind it?


Good question. However, the answer is in the if of your question.

A propellor does work - it pulls (or sometimes pushes) the aircraft through the air. In order to move the aircraft at a useful speed (or even at all) it has to spin pretty fast. 

The faster it spins, the more work it has to do. That work requires power. If the engine doesn't have enough power, it can't spin the propellor fast enough.

So yes, though the power and the speed of the spin are not the same thing, you won't get the spin without the power.

If you consider a car, the engine has to spin the wheels fast enough to move it along at the desired speed. And what makes it possible for it to spin the wheels fast enough is power. Otherwise, the wheels won't turn fast enough (or at all), and the car won't move fast enough (or at all). It's the same in the case of an aeroplane.