Understanding e-bike power/torque In the UK e-bikes are currently limited to a maximum of 250W assistance if you wish to use them on public roads. However, it seems that the motors do vary in terms of the available torque - from around 40Nm on some of the lighter e-bikes to 80Nm on those that are described as more "powerful". 

Now clearly they are not more powerful as they putting out the same 250W max, but what I don't quite understand is what difference that extra torque makes.

I'm used to measuring my power output in watts and thought there was a linear relationship between power and speed, so I don't quite understand where torque comes into it, and what difference this makes in practise.
Very quick answer - the torque numbers that are quoted are maximum torque values which do not correspond to maximum power.

Power = torque × rotational speed, so, at slow speeds the motor unit can provide more torque (and hence more acceleration) while staying within the power limit.

A bit more:

There a decent page here on the characteristics of DC motors. The essential takeaway is that DC motors can provide less torque the faster they spin.

Re:


  [I] thought there was a linear relationship between power and speed.


The power at constant velocity is that needed to balance the forces resisting forward motion - rolling resistance and aerodynamic drag for a bicycle or other vehicle. Those resistance forces generally increase with the cube of velocity.

If you want to play with the numbers there is an interactive calculator here.