Hi, Nora. First things first: please tell your mom that she has a fan in me. She sounds like someone who lives to prove that age is just a number!
In your case, I'm going to give the slight edge to
Disney's Pop Century Resort. While both of the resorts you mentioned are lovely,
Disney's Caribbean Beach Resort is a sprawling property and those who aren't as spry as they once were might find it challenging to cover so much ground on foot. On the contrary, I think you'll find that the main building (where you'll check-in and find the
food court) at Disney's Pop Century Resort is more centrally-located and convenient to most resort rooms.
Both of the resort hotels you are considering do provide unique transportation to Disney's Hollywood Studios and Epcot aboard the
Disney Skyliner gondolas.
Complimentary transportation by bus is also available at both Disney's Pop Century and Disney's Caribbean Beach Resort, too.
The one potential drawback to a stay at Disney's Pop Century that might factor into your decision is the higher likelihood of encountering large groups traveling together. Disney's value resorts often appeal to teams and other groups of kids. If that is a concern for you, booking your stay at Disney's Caribbean Beach Resort might be a better option.
I don't doubt that your family will have an unforgettable vacation no matter which resort you choose. I hope, though, that I've helped make the decision just a bit easier for you.