They're less accountable than LEA schools - for instance, not subject to FOI Act, don't publish exclusion rates, don't need to stick to healthy meals guidelines.
Schools commission their own services sounds good in theory, but means that those requiring more specialist services (eg translation, anything SEN related) have to fork out of their own budget which is same as schools with fewer pupils needing their services. Thus schools which need more support get progressively worse off. This may have been countered by the pupil premium, but don't know to what extent.
On a similar note, lots of schools commissioning same services means no possibility of economies of scale - eg paying consultant rates to a specialist to come in an advise on aspect of the curriculum whereas LEA would have someone in-house or be able to broker a borough-wide deal.
Sponsors get to run a school, including deciding large chunks of the curriculum, for next to nothing - when they first started, it was something like £4m, which is peanuts compared to the actual cost of running a school.
The large building projects are often funded by expensive PFI deals.
As others have mentioned, the staff often get shafted.
So basically, it might be good for your individual school, but has the potential to bugger up the system overall.