They're surprisingly different.
"Eminent" is usually applied to people. It is a somewhat formal, dignified, and rather old-fashioned word. It is often applied to people in a profession: the "eminent professor," "eminent doctor," "eminent statesman," etc. "Walter Lippman was an eminent journalist." "The eminent conductor, Leopold Stokowski, was famous for his orchestral arrangements of pieces by Bach."
"Preeminent" means "well above all others," "head-and-shoulders above others," "in a class of their own." It is stronger than just saying "best." It means "easily the best." It means better than the rest by a large amount. It can be applied to people or to institutions. "Rhode Island School of Design is the preeminent design school in the United States." "Jane Goodall is the preeminent authority on chimpanzees."