Answer:
There are quite a few things that would be different!
For starters, each state would act as more of its own country rather than just a state. They would likely have their own unique monetary system and you would have to exchange currency whenever you wanted to move to a different state for either permanent residence or a simple visit. It would be extremely difficult and troublesome for banks to keep up with all the different types of currency from ALL 50 states.
I would suspect there'd be enormous amounts of disputes between them, too. For example, many criminals sneak out of state to avoid being punished for a crime committed in another state. However, the laws could be entirely different in the state they fled to, and if the state had federal-level power, they could refuse to hand over the person to the other state because they are THEIR citizen now; they have no obligation to go through with the criminals' capture unlike how it is now. If you know the quarreling between countries in Europe such as Britain and France with the Hundreds' Year War, then picture that but 50 countries.
Congress wouldn't exactly be congress more than it would be something like a United Nations meeting, and the same goes for the Senate. On the note of government officials, voting would also be extremely different, as counting votes wouldn't be as time-consuming and each state would vote for its own presidents. I also suspect that each state would have its own party system as well :)
just a few things off the top of my head
Hope this helps :)