Literally does not matter. Consoles can't even reach 60fps most of the time, so they won't even be taking full advantage of 60Hz. Movies are 24fps, and most television tops out at 30fps, so there's not a huge reason to have a higher refresh rate unless you're using a high end PC.
I recently gave anime a shot as well, after hearing a lot of hype for Attack on Titan, and feeling like I'd exhausted every TV show and movie on Netflix. I honestly despised Attack on Titan, but my interest was piqued so I downloaded Cowboy Bebop, Code Geass, and Gurren Lagaan... and now I'm hooked.
Any anime recommendations? I have a massive list of things to check out now. I've probably added like 10 shows to my Netflix list, and downloaded a few others (FLCL, Death Note, Samurai Champloo, FMA, etc).
@bigboss1911: This wouldn't cause the significant frame issues you're seeing. It would just cause the RAM to run at 1333MHz instead of 1600MHz or 1600MHz instead of a higher frequency. Take everyone's advice and RMA your components.