Earlier translations of the Qur'an and other Islamic texts rendered the Arabic term "RASUL" as "Apostle" in the English language. More recently, Muslims seem to prefer the term "Messenger" instead.
Apostle comes from the Greek and just means "one who is sent", a "sent one". Obviously, it is implied that this person is sent with a task, either a message to bring or an action to perform.
The word "messenger" means basically the same, but stresses that the person is carrying a message (and implicitly it is assumed that it is not his own message, but one given to the person), i.e. emphasizes the content instead of the source.
The question arises, whether the Arabic title "rasul" more closely corresponds to the English term "apostle" or "messenger".
"Rasul" has exactly the same meaning as the Greek term "Apostle". The word rasul is derived from the verb "arsala" meaning "to send". Rasul is "the sent one" or "the one who was sent" or "the envoy".
Hatib bin Bal'a was the "rasul" of Muhammad to the governor of Egypt inviting him to follow Islam. The Kings and Khalifs used to call those whom they send to other kings or to heads of state a "rasul" to carry a certain message or probably a gift from them. Nowadays, heads of governments in the Arabic world use the word "mab'ooth" instead of rasul, derived from the word "ba'atha" which basically means the same thing as "rasul". ("mab'ooth khas" means: "a special envoy").
Just as in the Arabic world Muslims now want to reserve the term "rasul" for one person only, i.e. Muhammad, so the switch from "apostle" to "messenger" as the English translation of the Arabic title "rasul" is motivated by the desire to distinguish. Because Muslims want to avoid the impression that those who are commonly called "Apostles", namely the Apostles of Jesus Christ, are on a level of similar authority as Muhammad, the Apostle of Allah, the unique and incomparable, they want to distinguish Muhammad from them by using a title which is not commonly used for the ambassadors of Christ.
Nevertheless, the basic meaning of both terms is the same, "apostle" remaining the semantically closer translation for "rasul" in comparison to "messenger".