Not really. Postdocs are on temporary contracts to do research guided by a professor (normally). They really don't have a lot of other responsibilities. This varies by professor, but my old PhD advisor used to say that it was the last chance you had to have no responsibility and still do great research.

Asst. profs (normally) are tenure-track, meaning they have to write grants and bring in money, set up labs, create their own research, manage PhD students, etc.

Again, all this varies by department and/or university but this is, in general, the responsibilities of each.