New Mishne Torah
Here's the rub. Kafih's edition is based exclusively on Yemenite manuscripts. Frankel's represents primarily European manuscripts and printed editions. Rabinovitch's version, with his commentary Yad Peshutah, follows the Oxford manuscript. As far as I am aware, only Shilat made use of Genizah fragments.
The Maimonidean presence in the Genizah is very strong. Thanks to the Fustat synagogue, we have many fragments in the Rambam's handwriting, and many other pages written by his close acquaintances. For examples, see here and here and here (by the way, the Arabic original of the Guide is here).
Avi Lifshitz, a student of Shilat, recently completed a Master's thesis at Bar Ilan University studying the significance of these fragments in the study of Mishne Torah. This is an important step, that must be followed by an appraisal of the relative worth of individual scribes and copies.
One small example, in shorthand. Isure Biah 21:9 - "U-bilvad...". Frankel's edition testifies that this phrase is missing from most manuscripts. It is also not to be found in a genizah fragment, BL Or. 10832/1, from the Gaster collection.