Technically speaking, he violated the Logan Act

The Logan Act (1 Stat. 613, 18 U.S.C. § 953, enacted January 30, 1799) is a United States federal law that criminalizes negotiation by unauthorized persons with foreign governments having a dispute with the United States. The intent behind the Act is to prevent unauthorized negotiations from undermining the government's position. Violation of the Logan Act is a felony.

Only two people have ever been indicted on charges of violating the Act, one in 1802 and the other in 1852. Neither was convicted of violating the Act.

The problem is that the Logan Act is never enforced. Few people are interested in negotiating with foreign governments for the US, let alone doing so on their own dime. More importantly, those foreign governments won't just negotiate with anyone. So if you're going to do this, you have to have some connection with the existing government. Jimmy Carter (39th US president) went to North Korea on his own in 1994 and negotiated a deal about nuclear weapons. Then-President Clinton accepted the deal.

So, to answer the question at hand, no, there's no practical difference between Biden and any of the other people who have violated this act in the last 150 years. Biden was acting outside official US channels to shape US policy. While it's possible Biden might get investigated, it's highly unlikely he will ever face any charges for negotiating with Iran, just as Kerry and Flynn faced no charges for their negotiations.

This answer is an adaptation of that by user Machavity to Is there a difference between Micheal Flynn's actions and John Kerry's actions, with regards to legality? used under CC-by-sa