Do we need special permissions to drive from state to state in the United States?

If there is any chance that i might want to leave the state then I ask when renting. The answer has always been yes. If you need to know in advance, go to the websites of some companies and look for a "contact us" facility.

There appears to be some variation in rules. When visiting a new state, I check. This is worth doing anyway as the rules may differ a lot from home. For example, it took me a long time to become comfortable turning right through a red light. Four way stops were very unfamiliar as well.

Additional

I did not respond to licences. I am not an expert here but I would be very surprised if your licence was valid in some states but not others. I would expect all or nothing. Let us know where your licence is issued and someone may be able to help. My licence is from the UK and has been happily accepted in many states.

The road rules differ very little because there's a lot of work at the Federal level to harmonize rules. Speed limits differ dramatically, but those are posted (in some cases, few and far between). Don't speed in Ohio. Don't mess with (litter in) Texas.

Things which tend to differ state-to-state

You do see odd differences. For instance California has very few passing zones on 2-lane roads. Some regions are in love with roundabouts. Some states, you make U-turns a lot; other states hardly at all.

Drive right, pass left: Only use the left lane to pass. You would never realize this rule if you only drive in the city. It's inapplicable on a road too busy for the rule to make sense. And it's universally ignored in California and cities. Elsewhere, you can get written up for it, especially in rural areas. Don't be the one guy driving in the left.

Traffic signals can be sequenced in unexpected ways. Sometimes opposite direction traffic goes at once, other times in sequence. Sometimes lefts happen first, other times last. At intersections with signaled left turn lanes, they have differing ways to tell you "turn when clear". Some flash yellow, some flash red (technically wrong) and some don't allow it. Some traffic lights take a very long time (3-5 minutes) to go through all their phases.

Some roads eliminate all left turns with "Michigan Lefts".

Right on Red (when clear/safe, and after a stop) is legal everywhere except where a sign prohibits it, and also New York City. You just have to know that. Also in a few states, some traffic signals display a red right-arrow -- when lit, this prohibits a turn-on-red that would otherwise be legal. Some traffic signals have a green right arrow, explicitly authorizing a right turn without stop.

An octagon sign is only used for a stop sign.

If your American experience is only certain urban centers, you may be surprised to see surface railroad crossings on heavy and fast lines. It's easy to come up on them much too fast - a common mistake is to T-bone the side of the train at night. A circle-shaped sign is only used for early warning of railroad tracks. A Saltire (X shape) is only used at the railroad tracks. You must slow, look, listen. Many crossings do not have warning devices other than these signs. A slow train is not safer: Half of all fatals happen with train speed below 5 mph.

Do the traffic rules change: yes.

Do you need a permit: no.

Can the rental agency place restrictions of their own: of course.

And that's all there is to it. Usually you're fine driving out of state with a rental car, but leaving it at a dropoff point in another state usually costs you a (sometimes quite high) fee to the rental agency who will have to transport it back to the place you got it from (which is what the fee is there to cover).