And practically speaking, there's no difference between the wired or wireless Genie minis; they both function the same.
I can say without reservation, however, that the wireless Genie mini is fantastic and has been working like a charm, through walls, etc. for over two years.
Other than that, other folks are similarly correct. It's good to have another receiver available on the same network because if the Genie goes down, so do any/all minis.
Regarding tuners (and how many shows can record at once), each Genie DVR is considered to have 5 tuners. If you have another HD DVR in your household, it is likely 2 tuners, so you could record or watch up to 7 programs at once in that type of setup.