I think the reason your question hasn't been answered is that there are more than one rare earth elements.
If you are asking which is the rare earth element that is most important for technology I'd say it is neodymium. Neodymium is used to make very small and strong magnets that have allowed us to miniaturize our gadgets.. But they're also used in much larger technology too like wind turbines and oil drills.
Erbium probably comes in a close second because of it use in optical fibers and lasers and my third place goes to Lanthanum which has helped make car batteries more efficient. A nickel-lanthanum hydride battery is approximately twice as efficient as the standard lead-acid car battery and this has made electric and hybrid cars a viable option.
The strange thing is that rare earth elements aren't rare!!