The crucial difference is that the E.U. can impose regulations on citizens of member states that have the force of law, and those laws override national law: https://en.wikipedia.org/wiki/Regulation_(European_Union). The U.S. has a similar principal (the Supremacy Clause), and in the long term that resulted in the gutting of states as sovereign entities.
It can't do that without the agreement of the states though. Directives are created at the direction of the European Council, made up of the heads of state of every EU member state.
US states don't have veto powers over federal legislation, but EU states do. Hence Britain never joined the Schengen Agreement and kept its own currency. We had the best of both worlds and threw it away.
