All footways are also paths: path is a broader category of ways that also includes, for example, cycleways and bridleways.
If a path is dedicated for pedestrian traffic, then you can use the more specific category “footway”.

No information about width, surface, urbanity, etc., can reliably be extracted from the choice of either path or footway. The tags were not originally defined to have any of those meanings, and there is no really consistent usage that would allow inferring that information. If you want to represent any knowledge of that kind, you should add additional tags instead of relying on the relatively fuzzy definition of highway values like “footway”.