An executive producer is a producer.
But she is also an executive. She assumes a leadership role, and responsibility for the business/resources. She is involved in business executive level decisions. (As opposed to the director, who is involved in creative executive level decisions.)
In Hollywood, she raised $100 million, and it is her responsibility to earn a return on that investment. (Since she pays the bills, she can be "final say" on the movie. However, if she uses "final say" on creative decisions whenever she wants, she will frustrate the director. (It's the director's job to direct the creative aspect of the film.)
Director: I want 1000 extras. Exec Producer: No, we don't have money for that. Director: I want the curtains to be red. Exec Producer: No, I like green better.
On a smaller scale, she is the voice of reason to the director's mad fantasies. (She needs to be someone the director trusts so he will weigh her advice carefully.) She has to use her persuasive skills to advise the director on a suitable course of action so the production can be completed within budget.
Also, she is responsible for business. If a crew member is about to spill red wine on white carpet, it is her job to say "No" because she is responsible to the location owner.
The director's job is to think "Will this make the movie more entertaining/artistic?" The producer's job is to think "Will this make the movie more profitable?" (Or smaller scale: Will this offend film festivals? Will this decision make more people want to watch the movie?)
A good producer doesn't make the video good, a good producer can make the video look expensive. When a video has "high production value" it means that it looks like it was produced with a big Hollywood budget. It means the producer pulled together highly skilled creative talent (who often cost a LOT of money) so the visuals are in focus, the audio is without noise, etc.
You don't need a PhD to budget and hire expensive creative talent. But when you hire a "production company"-- make sure that those producers hire a director who can handle your material.
You need a PhD to direct the creative talent so the video accurately conveys your message at the appropriate level of scientific background knowledge for your audience.