I'm with Zigster, in that the answer to the question seems to me to depend on the family, the schools involved (both private and state alternatives), thc child, and to some extent the local community.
If a family has a sufficiently high income that the cost of private school does not entail significant sacrifices (e.g. simply that holidays are scaled back or cars not updated as often), then the additional benfit of the private school dores not have to be very large to be 'worth it'. However, if the sacrifices are very large (e.g. risking losing a house, not leaving enough money for food), then there is almost no degree of additional benefit that makes that sacrifice 'worth it'.
Equally, there are schools where the difference between the state and private options is vast, and therefore the private option more 'worth it'. Faced with the choice between the worst school in my neighbouring town vs Winchester, then the private option is worth it. Faced with the choice between my local private secondary and the local comprehensive (which gets the better results of the two), then the private option is not.
Equally, if a private school is the only way of getting your child something that they need - a minority sport, say, or a particular subject, or simply the chance to be away from a specific peer group - then that might be worth more to you than if the child would in reality take advantage of exactly the same options in both schools (I mean, the private school may have a fantastic drama hall ... but if your child has no interest in or aptitude for drama, and is in fact a keen footballer while the private school does rugby, then the advantage of the private schools' additional facilities TO YOUR CHILD is nil).
Finally, the local environment plays a part. My town has, for example, outsatiding community facilities for many sports, for drama and also for dance - e.g. for the latter, gifted childrendo not use 'in school' facilities but take classes in the community dance schools. In other areas of the country - for example where we used to live - such facilities are ONLY available within private schools.