Basically if you do a lot of light gaming or use DRM Video streaming Apps like Netflix, HBO, etc in addition to Kodi, then the Android TV OS versions of those on the Shield will be a lot easier to use than using a mouse and browser on a TV screen. But you will get better than 480p resolution for Amazon Instant Video using a browser than using the App on the Shield.
Compare Video Streamers
HERE
Concerning video, you will get properly synced Fractional Display mode (23.976/29.97/59.94fps) output from the Shield that you cannot get with any Mac due to an OSX limitation. However run Windows on a Mac (Intel Haswell and above) with Bootcamp and these are available. If you are not video judder sensitive this won't make a difference.
Both the Shield and Mini will only give you 8-bit video output using Kodi. The Dual Core i5 Mini should have the grunt to Software decode up to low bitrate 2160p 10-bit HEVC video, HEVC Hardware decoding on the Shield. Your Mini is a Mid 2011 Intel i5 I believe, I've tested low/mid bitrate 2160p 10-bit decoding on my Quad Core Mid 2011 i5 iMac. Ultimately 2160p HEVC Hardware decoding is better and preferred.
Not sure about 4K/2160p video output. You would need a Thunderbolt to HDMI adapter .
EDIT: looks like the 2011 Mini is limited to a resolution of
2560x1600 over that spec of Thunderbolt.
Better deinterlacing video quality out of the Mini using Deinterlace = Yadif, although I believe running a recent SPMC version may fix this on the Shield.
7.1 HD Audio Passthrough on the Shield. 7.1 HD Audio decoded to Lossless Multichannel Audio output on the Mac Mini. OSX Limitation. Really no Audio quality difference. 7.1 HD Audio Passthrough again if running Windows on the Mini. The BT Remote with the Shield has an Audio output jack I believe for private A/V use.
Unless you need Android Apps or are having major Kodi GUI usability issues with the remote you have I do not see a lot of advantage to upgrading.
Need Mid/High bitrate 2160p HEVC decoding I would get a Shield. Not a lot of 2160p content around at the moment apart from Test clips and 4K/2160p Netflix and Amazon Prime Video.