US citizens can visit Iran—at Iranian discretion—with a state minder the same way US citizens can travel to North Korea. You can't travel freely and you can't travel without personal scrutiny ('extreme vetting') from Iranian officials.
TL;DR: Visa applications take longer (but still usually <30 days) and you must be accompanied by a guide at all times, but other than that you're free to travel around.
It's worth noting that this is considerably looser than North Korea, where you not only have a guide, but can only go to the places that have been agreed in advance and your guides permit you to go to.