The processor on the Galaxy 7" is comparable to the Mini (not the same chip though). The Mini has a bigger screen, but Galaxy has more pixel per inch (pixel density) and about half the price of the Mini.
Just a personal preference, I have an iPad but want a 7" device to play with and bought the Galaxy 7" because of the higher pixel density and cheaper price. I will use it mainly to play video/TV.
Nexus 7 has a faster processor, but no slot for SD card. Many also reported problem with the screen that need to be returned/replaced.
So it is a personal preference, Nexus 7 is a winner in speed, and Galaxy 7" can expand memory easily via MicroSD cards.
Thanks, I have an iPad 3rd gen... and wanted something smaller... maybe for presents.