In a practical sense, there needs not be an operational difference, and is subject to scrutiny from the IRS to determine whether an organization is eligible non profit status
It honestly doesn’t matter what the tax statuses of either of the corporations are. If Musk had invested in OpenAI with the goal of making tons of money off their IP (as opposed to wanting to open source it) and then the board decided to just hand over all the IP to another corporation essentially for free, Musk would be just as validated in suing.