Despite the fact you've not responded to my inquiry, I think I may have got to the bottom of this issue, although it's not really a satisfactory solution.
When I close MyMoney using the 'Quit MyMoney' from the 'My Money' menu, it won't backup, nor does it when I use the 'Exit' option from the 'File' menu.
It does, however, if I use the little red dot, with a small cross, at the top left hand corner of the My Money window.
This isn't the standard method of closing down software on a Mac, it just closes the active window. This is normally a Microsoft method of closing a program.
I don't know if this is design or accident, but I thought you'd like to know.
Peter