The question isn't very clear. The difference between using which two things? "Their bodies' needs" is not a mistake. It means the needs of their bodies. Is this what you wrote or what the official transcript said?

"Every person" is singular, so the correct way to say it is "every person needs to consider his/her/their body's needs" "Their" is acceptable because it is sometimes used as a substitute for "his or her" but it is still singular because of "every person". "Their bodies' needs" is not always a mistake but it is a mistake in this particular instance. The thing that makes it a mistake is that the transcript says "every person". That means you are talking about just one person (at a time). Since one person has only one body, you cannot say "bodies". If you said "all people" instead of "every person" the analysis would change.