I went through major depression around 15 years ago and did a tiny bit of CBT with a counsellor who told me I was exactly the sort of person it wouldn't work on. It's about challenging your depressive thought patterns so something like, Original thought: "I can't do that, I'm so stupid." Challenge it with: "why do I feel that way? I've done this sort of task x times, I CAN do it." Sounds good in theory except someone like me would add an extra bit at the end, "why do I think I can't do anything when I obviously can, I'm such a freak for thinking these thoughts, why can't I just stop thinking this way." And on the chatter would go all day.

I haven't experienced mindfulness in a counselling situation or when I've been depressed. I've only read a couple of books about Buddhism but from the little I've read, it suits me much better. In the broader context of Buddhism, it says that life is about change. Rather than worrying about why change is happening, we accept it and live in the moment a lot more. So mindfulness could have helped calm my mind through my depressive episode. So taking my original example: Original thought, "I can't do that, I'm so stupid," Mindfulness: "you are feeling stupid right now but this feeling will pass. Everything changes." So it doesn't allow the brain to engage in a battle between 'good' thoughts and 'bad' thoughts.

That's only my interpretation from the little I know of both approaches.