A glib answer is: all sorts of things, the beauty of which is that it's their business.
A better answer is: special needs provision, longer hours and boarding school that make demanding careers possible, in-house extra-curricular activities (that state school parents also buy VAT-free).
And of course confidence they're receiving a good education, in the same way millions of families pay £1000s for a catchment area and VAT-free tutoring. For the overwhelming majority it's not "I want advantage / opportunity at the expense of others".
Most of us would be delighted to see free taxpayer-funded education improve in order to give us more / better choices, and because we're selfish enough to understand that when our neighbours / colleagues kids flourish in state education it's terrific for us...and conversely it's a tragedy when they don't.
Meanwhile knowing we're the strongest possible supporters of state education. We support state education much more than most of the people using it or advocating for its expansion; we pay, often several times over, for the non-existent unfunded place we're entitled to but don't use; we pay for other people's places; and the schools themselves contribute £5.1bn in tax on payroll, purchasing and impact on the downstream economy.
The irony, of course, is that a very few families are looking for the advantage/exclusivity/snobbery you mention, and they find it at some very few schools, which you'll have heard of. This tax will only serve to give them even more exclusivity than they already have.