There is a significant difference between what the three witnesses experienced and what the eight witnesses saw. The scripture says that the three witnesses shall behold it, by the power of God. This means that rather than Joseph Smith just showing them the plates, they were given the privilege of seeing Moroni, of hearing a voice from heaven proclaim the truth of the record, of beholding the Liahona, sword of Laban, the Urim and Thummim, and the breastplate (see commentary on how the 3 witnesses received their witness). The eight witnesses, in contrast, saw the plates, but the witness was not given by the power of God. Rather it was given according to the will of God (v. 13).