The "match version" is the so-called "authentic / player issue jersey" whereas the "stadium version" is usually referred to as the "replica jersey". I did a long post highlighting the general differences in this very thread, you can search for it.
Here are the key differences, in a nutshell:
- Authentic jerseys are slim-cut, about one size down compared to replica jerseys
- Authentic jerseys feature the laser-cut holes on the side panel, supposedly to ventilate sweat (a feature not seen on the replica jerseys). However no concrete, not even anecdotal evidence has been provided showing it is cooler than the replica jerseys in this regard
- The 14/15 authentic jerseys are made of the same fabric that you saw on the jerseys of the last several season, polyester made of recycled plastic bottles. The back panel of the jersey is made of mesh fabric which helps ventilate (a feature not seen on the replica jerseys)
- The 14/15 replica jerseys are made of a different fabric throughout the body of the jersey, different from the fabric they use for the authentic jerseys, more cotton-ish, to me it felt pretty comfortable
- The 14/15 authentic jerseys feature a higher-quality, bigger embroidered FCB crest whereas the replica jerseys feature a slightly smaller, average-quality embroidered FCB (the same crest we had on the training jersey)
- Color-wise I compared the two, no noticeable differences
That's about all I can think of right now. If you want the pro, on-the-field feel, get the authentic version, otherwise the replica version will suffice. Again, keep in mind that the size of the authentic jerseys is about one size smaller.
I personally ordered an authentic jersey.