The power [Watts] and torque [Nm] thing ought to be easily understood.
Power brings in the time element, how fast or how slow doing something is achieved.
Doing the task at all is where torque is involved.
Agricultural tractors up till the latest fast ones had small powered engines, but could drag a plough needing lots of torque, that being available because that power was deployed at a very low speed.
So returning to our bikes with the 250 nominal Watts, they can have whatever torque a designer chooses, but with that 250 Watts there comes a limited speed for that torque.
Power is torque multiplied by speed, as simple a relationship as that.
A significant argument for CDs in the pedelec case, is the assist is applied at the same point as the pedaller, so the bike's variable gear options are available to both the motor and the human. This ought to be an obvious advantage where a wide range of torques would be encountered, hills and some off road use, just as with a conventional bike we prefer to have a geared bike for hills. The torque at the wheel contribution from a hub motor, unless it features variable gearing, will always be limited to one maximum value.
In my case a CD enables the use of hub gearing and of a belt drive, way my preferred option for my particular needs. Fundamentally, I need a geared system, ideally a low maintenance one, and importantly as the bike goes in the car and house as clean a bike as possible. A front hub would IMO make a very poor functional alternative, a rear hub, clearly a non starter for my needs.
Force is measured in Newtons. Unless the driving force is greater than the retarding force the device stalls...or gets driven backwards.
Distance is measured in metres
Work is measured in Joules and is the product of Force x Distance.
The Work done in a second is called Power and measured in Watts
For motion in a circle the same principles hold.
Torque is the turning force theoretically available at an arm 1 metre long. from a pivot point. Hence Newton metre.
Distance is circumference of a circle or 2 pi R .
Work done in one rotation is 2pi R x Torque
Power is Work over time hence if that rotation is done in one second that is the power in Watts
All these are definitions and have no particular relevance to electric motors and are equally valid for any rotating machine.
Using gears if different diameters, the Torque and speed can be altered, while still consuming the same power.
There is an interesting point about electrical machines. While they are usually very efficient in converting electric power into mechanical rotating shaft power, ..figures of 90% not uncommon, they can be stalled . If stalled, they are still consuming electrical power, but not doing any mechanical work. And hence have zero efficiency. Under stall all the electrical power is being converted into heat, either in the motor windings or in the controller or in the battery. Likewise they are very poor efficiency when operating very far down their operating range.
The advantage of a crank motor is that the gears can be adjusted for different speeds, slopes and loads . A hub motor has typically only one geared speed for optimum efficiency. However as Woosh states, that is often enough.
