AP and DE are just two out of several ways to show rigor in your schedule. IB is another, but there are gifted/talented programs and schools running an all-honors curriculum that are also very rigorous.
AP is a known commodity, and colleges will usually tell you up front if those credits will transfer or not. DE credits may or may not transfer. AP grades stay on your high school transcript, but a poor grade in a DE class will come back to haunt your graduate school applications.
DE classes are appropriate if your high school doesn’t offer the subjects, if there are obvious scheduling conflicts, or if you have exhausted the course offerings in a certain department at your high school - or need flexibility in an otherwise homeschooled schedule. Beware of taking DE’s to avoid high school, though. College won’t like it if you appear - as some kids in a rush to get high school over with - impatient or egotistical, and they want to see that you can “play well with others”.