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The Sea and the ocean are the same thing - a large body of water not surrounded by land (c.f. lake).
Geographically, there is only one ocean that surrounds all the land on Earth. However, this global ocean is divided up and given proper names, generally five: the Atlantic Ocean, the Pacific Ocean, the Indian Ocean, the Southern Ocean, and the Artic Ocean.
The sea can also refer to this global ocean, but when used as a proper noun, there are a lot more Seas than Oceans. Many of these are part of a named Ocean; for example, the North Sea is the part of the Atlantic between Britain and Northern Europe. These oceanic seas are generally distinctive in some way - mostly surrounded by land, having distinct currents, or being the area affected by major river discharges. Sometimes they're not called seas, particularly when they are mostly surround by land: Gulf, Bay, and Channel are sometimes used instead and these can also be used of smaller bodies of water that would not be considered seas. There can also be seas inside other seas like the Adriatic inside the Metteteranian.
Other seas are not part of named oceans: for example, the Black Sea or the Caspian Sea - the latter is technically a lake as it is not connected to the global ocean at all.
Because of this usage a sea has the connotation of being smaller than an ocean. When used metaphorically "A life on the ocean wave" or "The cruel sea" the words are interchangeable.
However, when used of the specific, i.e. as a shorthand for a specific named bit of water, the sea and the ocean might refer to same thing: that bit of water over there. Which leads us to why there might be a different usage in Britain and the USA.
The names for the bodies of water around Great Britain are (clockwise) the North Sea, the English Channel, the Irish Sea, and the Atlantic Ocean. However, the first three are also part of the Atlantic Ocean. They are also the three that touch the England part of Great Britain - which is where English comes from. So, it's natural for someone from the UK to refer to the water as "the sea"
The bodies of water around the United States are (clockwise) the Atlantic Ocean, the Gulf of Mexico (considered part of the Atlantic), and the Pacific Ocean. So its natural for a North American English speaker to refer to the water as "the ocean" (or "the Gulf" - always capitalised - if they are from a state that borders it).
English often has words like these that are synonyms - except when they aren't. Because English is a mongrel language that swallows up words that other languages leave lying around.
Sea is a Middle English word which means it came from its Germanic roots and has cognates in Dutch (zee) and German (See). Ocean came in from middle French from Latin which got it from Greek which probably got it from a pre-Greek language, it has a modern French cognate (océan).
So, you have Saxons peasants saying "se" and Norman overlords saying "occean" to mean the same thing - the salty water around England. Over time, as people made maps, the big bodies of water got the more prestigious Ocean and the lesser bodies got the peasant's Sea.