The most important difference in my experience has been that pyright is a lot more rigorous and correct than mypy. I found that it was easy to confuse mypy, leaving large amounts of code unexpectedly not type checked. Put simply, I trust pyright and I do not trust mypy.
You also have a lot more control with pyright over where unknown types are permitted or what other checks should be performed. Also, it can scan library source code to infer types which I don't think mypy supports.