I'm no expert, but I do seem to remember in the one report I read about neem (Azadirachta indica) trees, the plants around them were still eaten by insects. Plus, all that foliage will negatively effect the amount of sunlight your veggies will receive, so I'd stick with the spray.

While the people in areas of the world where this tree is native have used all parts of the tree for many things for eons the pesticide potential is greatest in the seeds and that needs to be extracted so simply growing the tree probably will not do what you expect it to do.