Let's first assume that we are talking only degreed engineers, not "engineering technology" majors, railroad engineers, custodians, or anything else that gets that word applied. We will stick with accredited engineering curriculum, with a degree granted at the bachelors level or higher.
Note that each field/specialty treats it a bit different.
Traditionally there was a limited set of fields, in recent history, they would be chemical, electrical, mechanical and civil engineering. More recently, a lot of specialties have broken out. Some that come to mind include computer engineering (which has essentially become its own field spun off of EE), environmental engineering (cross of civil and chemical), biomedical (cross of mechanical and chemical or of pharmacy and chemical), petroleum (really a specialization of chemical engineering), nuclear (chemical or mechanical with an emphasis on physics), etc. I'm sure I missed some and mistepresented others, but I think you'll get the drift.
I'm a chemical engineer with a PhD which was based upon the study of atomic structures under chemical reactions using synchrotron radiation as a tool to perform in situ x-Ray absorption spectroscopy to study said structure and reactions. My day job is research-oriented, so can support the use of such techniques as a means of understanding fundamentals, but I very much emphasize using the fundamentals to build practical and functional systems and demonstrate real stuff. That dissociated me from academia, which I leverage, which focuses on ever smaller and more specific phenomena in general. I cross disciplines too (which is common). Most of my work is electrical related, so I'm effectively a self-taught EE.
For me, a PhD was more valuable (and more difficult) than a PE. Though some Chem E friends have PE licenses, the majority of folks I know with PE licenses are EE, MechE or Civil. Why? Because this means something in projects that affect the public good, specific peoples' property, etc. As an example, say I, as a chemical engineer, was designing a refinery. A pressure vessel could explode and create major problems in the local area. I might specify the process and requirements, but a mechanical PE would probably work for a company making tanks, and "sign off" on the design.
So an engineer I'd say is someone who has at least completed a Batchelors degree from an accredited school. A PE is a term that is say is analogous to a CPA. You can have an accounting degree, you can probably even be granted a Doctorate of Philosophy in accounting. To do public work as a primary job, you would then get a CPA license. That's probably the best analog. You can be a book smart and horrible accountant (engineer), or you can have "not taken the test" and have great intuition.
A fraction of engineers go into medicine (due to the fact that most of the body's processes can be beautifully defined in engineering terms, which is something that doctors do not intrinsically learn/get), finance (due to math and reasoning skills) and law (due to technical and thought process/mindset, plus substance of the preliminary education vs "pre law", "poly sci" and other b s majors).
Depending upon the field and Position description, I wouldn't hire a PE over a regularly degreed engineer, but like everything, there are other factors that come into play.
