I'd say since you're already using a Mac, it makes more sense to be on the same platform, but that's just me.
I'm not familiar with the X51, but I'd say the mini is a great computer (I'm using a 2009 mini right now)
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A quick google seems to show the X51 to be a top notch compact computer that has superior specs to that of the mini. If you want a computer to do gaming then that will be the better choice.
If you want something on the same platform as a back up then perhaps the X51 is a better choice. I'm not sure how demanding a mine craft server is, or if a windows implementation would be better but the gaming rig looks beefy
