There is a difference for Mayo MN for research-intensive postgraduate, and there is a definite advantage if applying for East Coast or certain institutions in the Midwest due to past experience. The 2+2 program will have decent opportunities but does not have the same level of history as the MN side. Really no negatives, but there are a couple of situational areas where MN has better connections.
I'll give a concrete example. If you're applying to NIH/NCI for some of the more arcane ones (Medical (Biochemical) Genetics, Heme/Onc, Path, or AI) which there are significant research links, MN is probably going to have an edge over AZ or FL especially from the MD/PhD routes simply on faculty recognizance between the programs if not outright spending summers there beforehand. But for almost any standard residency through ERAS, Mayo is just Mayo.
But with the above, it is really more what the student does in the program. A MN grad that does not have a side business in one of the other buildings is probably just as competitive. The selection though is such that it is unlikely that the student has no professional extracurriculars.
And Emory is just fine. Either/or are good choices.