There are different frame rates for MP4 and .MOV videos, not all of which are supported by the Videos app - those that are supported are listed on this page :
Video formats supported: H.264 video up to 1080p, 30 frames per second, High Profile level 4.1 with AAC-LC audio up to 160 Kbps, 48kHz, stereo audio in .m4v, .mp4, and .mov file formats; MPEG‑4 video up to 2.5 Mbps, 640 by 480 pixels, 30 frames per second, Simple Profile with AAC-LC audio up to 160 Kbps per channel, 48kHz, stereo audio in .m4v, .mp4, and .mov file formats; Motion JPEG (M-JPEG) up to 35 Mbps, 1280 by 720 pixels, 30 frames per second, audio in ulaw, PCM stereo audio in .avi file format
If you've got them into your computer's iTunes then you could try selecting them in the Movies part of your iTunes Library and doing Fie > Create New Version > Create iPad Version and then try syncing those versions to your Mini.
Is there any difference between the video supported on an iPad mini vs. an iPad (iPad 3)?
Until today, I would have assumed of course not...but I've got 2 iPad Minis for my kids and an iPad for myself. There are a series of videos in my iTunes that play fine on ATV and my iPad but when transferred to either of my iPad Mini's don't work.
When I sync to either of my iPad mini's, I see the movie being transferred over. The mini has less storage space as a result of the transfer as well so the content / data is on the Mini somewhere - but the Video App does not see the video (either by going to the App or going to Usage in system preferences to see what's in the video app).
So the summary is :
-- 2 X iPad Mini's and an iPad 3 syncing to the same iPad library
-- Movie A works on iPad 3, but is not recognized by the Video App at all on either of the Mini's
-- I've tested this on a half dozen different movies, all were created / formatted for iPhone / iPad - all work on iPad 3, those that don't work fail on both Mini's