OP, I agree there are generally vast differences in the education and experiences offered.
If you take any child and put them in the local comp that child will likely perform worse and have worse outcomes than they would in a private school. This is supported by many universities who exclude privately educated young people from their widening participation programmes and actively seek young people from schools where results are worse.
If you feel like this though, show it and talk about with your daughter all you'll do is make her less confident and conscious of the differences. You can control this- that is in your power.
You can also look at local am dram groups for her which will help with confidence and public speaking. You could support any interests that she has, get her to take part in after school activities. Take her to museums, the library, art galleries, parks, theatre, scouts, cadets anywhere you can and everything you can so she has as much variety and as many experiences as possible. Talk to her about them.
Encourage her academics, make sure she has everything she needs, does her homework and expand on ut if you can. If she's a bit older sign her up to extra courses either online or in person. Try to instill in her why her education is vital.
Look at the local schools, is there a grammar nearby that she could apply to. Do any of the private schools offer bursaries or discounted rates?
I couldn't afford private for my children but one got into a grammar and the other has sen so he's in a independent specialist.
Take some control and look at what you can give her rather than what you can't.