Private preschool might be a good idea if:
Your child deserves a flexible and creatively controlled curriculum. Public schools have a lot of bureaucracy. Private preschool teachers have the flexibility to change lesson plans in response to cues from the children as they feel necessary. This amounts to a more personalized instruction and a more robust learning experience for your child.
Your child is too young for kindergarten but has mastered preschool material. Private school can move your child up a grade level, if needed or provide more in-depth instruction.
Your child learns through play and private preschools can determine exactly how much play each child needs.
Your child needs to learn and play with children that are like them. Private schools don't discriminate, but are more selective when it comes to choosing children to enroll.