The term prior restraint is used to describe administrative and judicial orders forbidding certain communications when issued in advance of the time that such communications are to occur.
Alexander v. United States.
A criminal statute restricting speech based on its content also forbids certain communications in advance of the time that such communications are to occur.
So what is the difference?
A criminal statute is made by the legislature, rather than the executive or judicial branch. That’s a distinction, but is it a difference?
Can the State constitutionally do with legislation something that it cannot, consonant with the First Amendment, do administratively? It seems unlikely—state action is state action.
The other distinction between prior restraints and criminal laws is that the former are generally aimed at particular speakers, and the latter are not. I can see an argument for that distinction making prior restraints more constitutionally suspect than content-based restrictions (because they chill speech more directly), and an argument for that distinction making criminal statutes more suspect than content-based restrictions (because they chill more people’s speech).
I’m not sure how that shakes out, but I have a theory that the nearest civil equivalent to a content-based criminal statute is a prior restraint.