The reality is that no. Saul is the Hellenized form of the Hebrew name Saul which means "Required, Requested." Paul in Latin (Paulus) means "little". Before his first missionary journey, Paul identified himself as Saul (Acts 7:58; 9: 1-27). By the way, the risen Christ called him by his Hebrew name, Saul, when speaking to him in the Hebrew tongue (Acts 26.14). During his missionary journey Saul begins to identify himself as Paul (Acts 13.9). Note that although Saul had an encounter with the risen Christ on the road to Damascus, Jesus never told him to change his name as it did with Abraham and Israel. In his letters, Saul always identified himself as Paul. Why? Because Paul is in Greek and Saul is in Hebrew. As Michael is Miguel in English.

What's the importance?

Although the most direct explanation is to change the name according to the audience to which it was addressed and where it was, there are those who indicate that this name change could also have been something a little deeper on the part of of the apostle.

It has been popularly assumed that Saul's name was changed when he became a follower of Jesus Christ, but that is not the case.[6][22] His Jewish name was "Saul" (Hebrew: שָׁאוּל‬, Modern Sha'ûl, Tiberian Šāʼûl, "asked for, prayed for, borrowed"), perhaps after the biblical King Saul, a fellow Benjamite and the first king of Israel. According to the Book of Acts, he inherited Roman citizenship from his father.[citation needed] As a Roman citizen, he also bore the Latin name of "Paul" – in biblical Greek: Παῦλος (Paulos),[23] and in Latin: Paulus.[24][Acts 16:37][22:25–28] It was typical for the Jews of that time to have two names, one Hebrew, the other Latin or Greek.[25][26][27]
Jesus called him "Saul, Saul"[28] in "the Hebrew tongue" in the book of Acts, when he had the vision which led to his conversion on the Road to Damascus.[29] Later, in a vision to Ananias of Damascus, "the Lord" referred to him as "Saul, of Tarsus".[7] When Ananias came to restore his sight, he called him "Brother Saul".[30]
In Acts 13:9, Saul is called "Paul" for the first time on the island of Cyprus–much later than the time of his conversion. The author (Luke) indicates that the names were interchangeable: "Saul, who also is called Paul." He thereafter refers to him as Paul, apparently Paul's preference since he is called Paul in all other Bible books where he is mentioned, including those that he authored. Adopting his Roman name was typical of Paul's missionary style. His method was to put people at their ease and to approach them with his message in a language and style to which they could relate, as in 1 Cor 9:19–23.[6]

WIKIPEDIA

.