As a verb, stay has several synonyms: remain, linger, tarry, sojourn, abide, wait, etc. As a verb, stop is related to arrest, check, halt, and terminate. As these parallel words suggest, stay and stop do differ in meaning. For example, one stays rather than stops at a motel because stay means "to remain," "to abide," and stop means "to terminate." However logical this distinction, Americans pay little attention to it: one stops or stays at a hotel or motel as he pleases. Stop is a more popular word than stay, as note such phrases as "stop over," "stop by," "stop off," and "stop in," to which has been added the noun stopover.
Dictionary of problem words and expressions. Harry Shaw. 1975.