Haven't heard the Bose soundlink mini but I have the soundfreaq sound kick and the Bose soundlink normal size and they both have their plus points.
The Bose is heavier and better built, the bass is much warmer, a little too warm for me at low volumes (why can't we have tone controls anymore!!). The treble response is there, and quite clean, but overpowered by the upper bass tones. I have a problem with mine losing charge when stored, too easily, already had one battery replaced under warranty.
The sound kick has less bass, it is there but much less warm. The treble is nice and crisp, maybe not so clean, but amazing for the 80 quid I paid for it. It is much lighter and more plasticky. I've noticed the bluetooth on mine needs to be reconnected every now and then, others reported it on the web as well, you just have to press the bluetooth pairing button and it will reconnect to the iPhone or ipad on first power up. Overall a much better price/value.
Would be interested to hear the difference to the Bose mini so will try and have a listen and report back