Between the stator windings and rotor is a small air gap. What would happen if this gap were larger than it is? Surely the magnetic field set up by energizing the stator windings would still cut the rotor and induce an emf across it. So what effect does this air gap have on the performance of the motor?
Air has a much higher reluctance (the magnetic equivalent of resistance) than the magnetic materials used in the motor. The smaller the air gap is, the lower that reluctance, and thus the higher the magnetic flux (which is the magnetic analog of current), allowing the motor to work more efficiently and at a higher power.
Smaller air gaps also minimize leakage flux, which means more of the flux that is produced actually does something, rather than just going through the air and doing nothing to help the motor work.
Let's say you design a motor to operate at a given field strength. Most of the magnetic path reluctance comes from the air-gap. Therefore the larger the air-gap, the more H field and therefore length of magnetic material you need to establish the field, so the more expensive the motor is to make. The size and weight also go up as the amount of magnetic material.
Some motors seem to have quite a large air-gap. That comes down to cost as well. A small air-gap needs precision machining, which becomes prohibitively expensive for very small gaps. There's an optimum size of gap for any size and target market of motor that minimises the total magnet and machining costs.
Note that although this is saying the same sort of thing as Hearth's answer, it comes from a different direction.