I've bought both of these and my GF thinks I just like buying gear, which is true, but I was trying to tell her the differences.  I thought I'd share some of my thoughts regarding the similarity and differences. 

Similarities:
both lightweight
both are simple to use
.1 micron filter 


Differences:
BeFree's flow rate is a lot faster. 2 liters per minute
Mini can filter 100,000 gallons vs BeFree's 264
Clean up is easier with BeFree just rinse it out, Mini can backflush
BeFree has a larger mouth (43mm) so filling it up is easier but is hard to find bag attachments.  Hydropak makes a large one.  Mini's mouth is smaller (28mm) but you can find lots of bottle attachments.
Mini comes with a straw, so in case you lose the bag or it gets damaged you can drink from the stream directly
Befree's filter fits inside the water reservoir, so you can pack it intact vs the Mini.


In summation, I told my GF is was necessary to buy the BeFree because it's great for day hikes- easy clean up, fast flow rate, easier to use.  I still use the Mini, I like the straw attachment and the longer life- I plan on using it for long hikes and backpacking.