I'm quite sure that the type of the pot influences the quality of the roots, some pots may be more adapt for some species, it depends on their necessities.
For example I know that clay pots allow the water to evaporate through the pot's walls and this may be good for the species that need to get dry quickly (and that don't need light).
For the thickness of the roots I also think that it is a matter of moisture, so it depends on the kind of the pot but also on how much do you water them.