Could be anything really, ranging from 'they didn't stick to the US ranking system because not the entire world revolves around the US' to a general 'who cares'. Neither of them have a military background, so it's also possible that they simply didn't know. If I can speak for myself, I think Commander sounds a little more formal than Captain. If I have to put that in logic, I'd say that Captains have ships, Commanders have units, and Admirals have fleets. So if I were to say it like that, I think it's to avoid confusion, since not a lot of people will be intimately familiar with the precise order of ranks, while almost everyone knows that a Captain has a final say on his ship.

Watching enough Star Trek, I'd say "yes, the Captain does give the orders on his/her ship."

Which specific thing in the show are you questioning? It's kind of vague.

It has been my experience that stories that are not straight-up military dramas by nature sometimes play fast and loose with ranking/terminology. To use one example, ZAFT, the colonies' militia from the Gundam Seed series, has "Captains" who are in command of ships, and "Commanders" who are in command of entire teams that consist of one or more ships, and thus outrank Captains.

Also, sometimes the same terms are used on different rank scales, which can confuse the issue. From my own admittedly limited experience, a Captain in the United States Marine Corps and a Captain in the United States Navy aren't the same thing - IIRC, a USMC Captain is an O-3, but a Captain in the USN is an O-6, equivalent to a USMC Colonel.

Bottom line: I don't think Bryke were really all that concerned about the FN military hierarchy matching up exactly to any particular real-world military hierarchy.

You're right about the differences in the rank of Captain between the Navy and the Marines.  In fact, the United States Army and the United States Air Force also use the same officer tier alongside the USMC:

O-1 = Second Lieutenant (2dLt)

O-2 = First Lieutenant (1stLt)

O-3 = Captain (Capt)

O-4 = Major (Maj)

O-5 = Lieutenant Colonel (Lt Col)

O-6 = Colonel (Col)

O-7 = Brigadier General (Brig Gen)

O-8 = Major General (Maj Gen)

O-9 = Lieutenant General (Lt Gen)

O-10 = General (Gen)

Also to note, Commander (outside of the Naval rank) is more of a duty title or position than a rank if we're talking about those above the Naval Captains.  Captains fall under the command of an Admiral who is the Commander for a Regiment or a Wing (I'm an AF vet, but I really don't know much about US Navy chain of commands).  For instance, on the AF side, a Lt Col is the Commander of a Squadron....for example: ###rd Training Squadron (TRS) or ###st Civil Engineering Squadron (CES), but he would also fall under the command of the Group Commander (a full-bird Colonel).

That was basically the whole reason why I was asking...