My dd went to a selective independent on a bursary; her brother went to a grammar (he's in year 10). Dd has now moved across to a different grammar for sixth form.
Pros of independent over ds' grammar were more time off curriculum (eg robotics day, Roman day etc) and more trips. Extra curriculars were good too and dd went to lots of different clubs, which also meant her years 7-9 were more varied and fun than ds'.
Having said that, she didn't find anything she actually stuck with! And ended up doing singing lessons outside of school anyway, as she wanted a more rock/pop approach and school singing lessons were very classical. So really- it's just a case of having a bit more fun and variety at school rather than anything life changing.
The time off curriculum/more variety benefit only really applied for years 7 to 9; after that years 10 and 11 were totally GCSE focused and she might as well have been at the grammar (one of the reasons for moving for sixth form too as that would have been the same).
Another pro was more individual attention - staff were more readily available at lunchtimes for drop in sessions and they were paired up with sixth form mentors for some subjects. However, my dd is very independent and not one to ever ask for help, so she really didn't benefit from this much! Well maybe she did in Maths, as she rather liked the male teacher...!
Her year group at the independent school was 90 (all girls) whereas it's 200 (mixed) at grammar. She felt 90 was nice and cosy, she knew everyone and felt very comfortable. Having said that though, ds knows everyone in his year (150) and hasn't had any social issues.
In terms of confidence - dd has no more confidence than any of her peers at grammar now! I suspect maybe it's a parenting thing?
Overall, I'm not sure I'd feel it was worth it for £13k a year, but for what we paid I have no quibbles.
On the downside, we found the independent much more pressured in years 10 and 11 - maybe that was partly because it was all girls though. The pressure came from her peers rather than the teachers.
She prefers being in a more 'normal' environment at grammar, as she felt the independent school was a bit of a 'bubble' and girls didn't really know how privileged they were.