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They are both essentially the same thing but a CPO is a previously leased car. It's really the only difference. And warranties vary in price depending on what your looking for, such as what you want covered and the amount of miles you want covered to. Most warranties add about $4k or so. And not sure if the warranty would be mini specific but any warranty should cover any car part replacement regardless of make or model.
I think that for a MINI a CPO is worth the premium over a regular used car. The cost and frequency of repairs will make the piece of mind worth it. Now on a Honda or Toyota, I would buy used over CPO any day to save over the mark-up price.