It's not necessarily that any branch is more "elite," it's more just different roles. Army and Marines overlap quite a bit these days, the main difference being that Marines focus a bit more on amphibious operations. Then the Air Force flies, Navy does water stuff.

Then you have different branches within each branch. I'll use the Army as an example since that's what I'm most familiar with. Generally the hierarchy goes SOF guys (these are your Green Berets, Rangers, etc) > Infantry > Other combat jobs (Engineers, Armor, etc) > Everyone else.

Both the Army and Marines have Infantrymen, and this is where most of the confusion, I think, comes from. Both are frontline troops trained to fight and kick in doors and do cool Infantry shit. Ultimately they're basically the same thing, but Marines in general have a much stronger, well-known "Warrior Culture" of sorts. Once a Marine, always a Marine. Every Marine is a rifleman. Semper Fi. All that shit. Whereas the Army doesn't have that, except in the Infantry, and to a lesser extent, other combat MOS's (MOS = your job in the military.)

Basically, Marines care a lot more about their image and the pride of being a Marine, and that shows in movies, recruiting ads, all that. Even though most jobs even in the Marine Corps aren't even combat related, it's that badass, tough, warrior image that makes them stand out. Whereas say the Army for example, you're only going to see that sort of pride and motivation in the cool guy jobs like Infantry and the SOF units. It's ultimately just a culture difference, which from an outsider's perspective looks like the Marines are more elite when that's not necessarily the case.