Would I notice much of a difference between a new 2014 Mac Mini and the 2012 server edition? I'm looking to replace my home server with a Mac Mini. I do a lot of conversions between movie formats (ie MKV to mp4) and was curious if it made more sense to go with the 2012 server model or the brand new (2014) model. 

I would be bumping up the specs of the new model to the 3.0GHz Dual-Core Intel Core i7 and 16GB ram. Likewise I would put 16 ram in the 2012 model

I suppose my question is would I see much of difference between these models for things such a streaming playback for 1080p to my appleTV, and converting movie formats?
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.