First (mathematical) use of the term cited by the OED: F.L. Griffin, An experiment in correlating freshman mathematics, MAA Monthly (1915), p. 328. Griffin later published a book, Introduction to Mathematical Analysis (1921), pp. 190f. His purpose in his article and book is the improvement of teaching. He also states that the notation is "common in scientific work" (1915), which seems the basis for calling it "scientific notation." One advantage: "This avoids operations with long rows of zeros or decimal places." (1921) He also argues that familiarity with the notation will aid the student's learning and appreciation of logarithms.
The broader notion that a scientific notation might allow the reduction of solving problems to symbolic operations, such as Vieta's algebra notation did for solving problems in numbers (see @Katz's answer), grew in the 19th century. By the late 19th century and on into the 20th, you find people arguing for and proposing scientific notations for phonetics, music, color, and the scientific teaching of English. At least one of these projects led to something still in use, namely the IPA.
The OP may be the first person interested in a comprehensive history of scientific notation. As a representation of numbers, I do not know of a work that examines the use of the notation "so common in scientific work" prior to 1915, according to Griffin. I know a random instance or two. Airy's Numerical Lunar Theory (1886) has some tables in which the entries are integers to be multiplied by a "unit" at the head of the column that is a power of ten. This is not the present-day convention for scientific notation, but it is similar. As far as the history after Griffin, I suspect scientific notation evolved due to the need for convenient forms of input/output for computers. I realize the OP is not looking to start research into the question, but assuming the basic research has not been done, I offer these "waypoints" in the hope they are helpful.