Honestly, the degree of independence is likely due to their history as much as anything else, and the fact that they operate completely different from the rest of the Navy.
As for why the services have overlap, this is usually with specialized forces.
For example, aircraft:
Army Aviation's combat aircraft are helicopters, primarily used to support ground forces (close air support and transport), while they have a number of reconnaissance and light transport aircraft (transporting individuals and small groups).
US Navy aircraft are primarily sea based (or capable of being based on ships), and transport that is required for them.
Marine aircraft are (like the Army) primarily close air support, transport, and reconnaissance, though more capable than the Army for air to air combat.
The Air Force having the land based bombers and fighters, heavy transports, previously the larger long range reconnaissance, etc...
If a new military was formed now, I suspect the elements would be more distinct. In fact in modern days I suspect the distinction between the branches themselves would be quite blurry at least.
The US armed forces (along with most other militaries) has had a long history of rivalry that goes beyond good natured ribbing. Everybody wanted everything they could have under their own control.
As for the "space marines" in the future, who knows, but I expect that if military capabilities were to spread beyond orbit the Air Force, not the Navy (like most fiction shows), would have official control of it until it grew and broke off, much as the Air Force was to the Army. Unless a fourth party is created right off the bat to stop any fighting over it.
Also: historically, I believe, the Marines weren't just the Navy's infantry, but the US ground presence overseas in general.
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