The educational assessment is very important, because it gives access to services and supports; but if you really want to know the full picture, it's worth seeing a private practitioner in parallel. The two processes are not equivalent.
jeimey, we are not dealing with ASD, but I want to second what DeeDee said - for any child struggling with a disability. I think that sometimes as parents it's very easy for us to get so caught up in the world of school - how to help at school, how to get around impacts at school, how to get services at school, how to make sure our very bright kids are successful at school - that we forget that although school is a large part of our children's lives, it's still only one piece. As parents we need to prepare our kids for *life*, and school is a subset of that - if we focus on the big picture, school falls into place (not without a lot of advocating - but the key is understanding that the disability impacts our kids across their life, not just at school. Having the correct diagnosis is really helpful - and sometimes very difficult to get to. Having the advice of private practitioners can also help with much more than just the diagnosis - they will be available for you (usually) for more than the eval - if a bump in the road crops up one or two or seven years down the road, you can re-consult with the same person who initially evaluated your child. As part of the initial evaluation, you will typically get their references and opinions on where to seek additional therapies or consultations that your child might need - you won't get either of these types of advice through school.
Best wishes,
polarbear