I would say that the distinction between the two groups is, roughly:

"Corporates" are the leaders of the many large corporations in this country. They want broad protections for corporate power against labour, corporate profits, lowered regulations, etc.

"Oligarchs" are the leaders of the few titanic corporations, like Bezos, Musk, and Zuckerberg. They want to become the zaibatsu of modern America, essentially being given total control over the economy divided up between them.

There are many things they want that are in common (for instance, the removal of regulations), but that prime desire of the oligarchs is directly at odds with the continued existence and livelihoods of the corporates.