How to describe a store supervisore, and what's the difference between a store manager and a store supervisor?

Manager and supervisor are fairly loosely defined terms.   However, a supervisor is always physically close to the team members.  Supervisors usually have a narrow range of specific duties, often concerned with ensuring that there are enough team members present to cover all roles, allocating individuals to particular locations, solving routine problems with equipment and overseeing the quality of work.  A manager may do all these things, but will tend to have other tasks as well, perhaps to do with planning or finance.

Without specific details it's hard to answer your exact question.  However, a store manager is the person in overall charge of a store.  A store supervisor may be in day-to-day charge of a store when a small group of two or three stores are managed together for the purposes of stock ordering, finance, etc.