Would I notice much of a difference between a new 2014 Mac Mini and the 2012 server edition?
The 2012 server is much more powerful than the gimped high end 2014 version and has beaten it in speed tests, too. 

You cannot install your own RAM in the 2014 model.

You would see no difference in regard to playback between the two, a MacBook Air i5 could handle that task just as ably. Where you will see the difference is with conversion times and processing power, with the 2012 server coming out far ahead.