Very little, it's mostly an FDA regulation nuance that you would never care about other than it allows them to use a slightly different official name, which they market as 'deluxe' because it obviously catches your attention.

Interestingly I see OP has already posted about milk rather than byproducts and oils. As far as I can tell, milk is always used in all their cheese. I am not even sure you can call anything a cheese without milk. Though the internet certainly is full of 'American cheese is evil' gut feelings.

American cheese is basically cheese + sodium citrate. Not the devils chemistry. And yes you then add all the normal stabilizers and shelf-life increasers to create a 'cheese product' rather than a regular wheel of cheese.

Or more simply, think of deli meat/cheese vs pre-packaged on the shelf. The deli is usually better quality and more expensive because it will go bad faster than packaged (not always, you can certainly find deli meat pre-packed too). Deluxe in this case would be buying from the deli, original would be whatever is on the shelf. And if you're thinking, that's not a huge difference, you are correct.

Super bonus info: Since American cheese is just cheese + sodium citrate, you may be wondering would this work with other cheeses? And yes, it does. You can make melty delicious cheese in any flavor you want. Imagine American cheeses goo factor + blue cheese on your burger? Sploosh.