What's special about 65mm/70mm film?

I read a lot of movie trivia on IMDB, and I sometimes come across movies that were shot on 65mm or 70mm film. This was apparently the norm for Hollywood movies in the 50s and 60s, but is almost never used anymore. Kenneth Branagh's 1996 version of Hamlet was the last Hollywood movie to use it until 2012, when Paul Thomas Anderson used it when making The Master.

My question is, what difference does this make? Is there something subtle in the visuals that only a trained filmmaker would be able to spot, or is it supposed to be noticeable by the average moviegoer? In the trivia page for Hamlet, it says that Branagh chose to use 65mm film because "certain shots could only be achieved in 65mm". What type of shots would those be?