Certainly all previous baptisms had been performed by authority from God, but perhaps we have in this prayer a new emphasis on the source of the authority. This is also perhaps a reason for a language difference between the Book of Mormon prayer and that found the Doctrine and Covenants.
“Some students have raised the question as to why the words of the baptismal prayer in the Book of Mormon differ slightly from the prayer listed in the Doctrine and Covenants. In this dispensation the Lord has counseled us to use these words in baptizing a person, after calling the candidate by name: “Having been commissioned of Jesus Christ, I baptize you in the name of the Father, and of the Son, and of the Holy Ghost. Amen.” (D&C 20:73.) The only difference in the two prayers is the introductory statement. In the Book of Mormon the disciples were counseled to say “having authority given me of Jesus Christ,” whereas in this dispensation we are told to say “having been commissioned of Jesus Christ.”
One possible explanation for this difference may be associated with the problem of translation. For example, it is not always possible to translate exact words into other languages; thus, the baptismal prayer is not exactly the same in all languages. Another possibility for explaining this difference is that the disciples in the Book of Mormon received their authority directly from Jesus Christ; therefore, they rightfully could say “having authority given me of Jesus Christ.” However, in this dispensation priesthood bearers have been given the power to baptize from John the Baptist, who was commissioned by Jesus Christ to come to earth and restore this authority. Therefore, in this dispensation we use the words "having been commissioned of Jesus Christ.” (Daniel H. Ludlow, A Companion to Your Study of the Book of Mormon [Salt Lake City: Deseret Book Co., 1976], 262.)