When I look at the label for Neem oil it says that it can be used for white flies, aphids and mites. I don’t see it labelled for the more common insects that do damage to apples- plum curculio, apple maggot, etc. I don’t think it would have any effect on them.
If you want to control insects organically you can either bag the apples or spray Surround which is a kaolin clay that acts as a fruit protectant. Or if your willing to use synthetics you can spray an insecticide to deal with the problem.
For an organic spray schedule and plenty of advice you can look here-
For a synthetic spray and more advice you can look here-