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All messengers(Rasul) are necessarily prophets(Nabi). However not all prophets are messengers. Messengers usually were sent to their respective tribe/people [the exception being Muhammad (peace and blessings be upon him), as he was sent to all humanity] with a set of revelations recorded into a form of scripture & comprising of guidance and divine law.
While prophets came to a tribe/people as warners and teachers of existing religion but were not given a scripture or divine law. This is very important distinction that should be known by every Muslim, because some sects and false-religions like Ahmadiya, Bahai & Submitters use this in opposite way as a point of deception.
The general position based on hadith narrations is that Allah subhana'wa'tala sent 124,000 Prophets among whom 315 were Messengers and the last Prophet & therefore the last Messenger both being Muhammad(peace and blessings be upon him).