Do children in private schools receive the same help as those in public schools?
It depends. Any student attending a school that receives public funds* is guaranteed a Free Appropriate Public Education (FAPE). For students with disabilities this means they may be eligible for services or accommodations under one of two laws - the Individuals with Disabilities Education Act (IDEA) or Section 504. IDEA is what is commonly known as "special education" whereas Section 504 is a Civil Rights law that covers children in public school settings. Both of these laws provide appropriate accommodations to eligible students with disabilities.
Unfortunately, if a private school does not receive any public funding and is not wiling to provide accommodations or services voluntarily, then there is little you can do legally, as these laws do not apply to them. However, some private schools will be willing to implement accommodations to help students with disabilities.
* Public funds sometimes slip into private schools through subsidized school lunches or busing programs.