There is very little difference. Both come from the Latin word for 'sea' which is mare, maris. Marine is an adjective meaning 'of or pertaining to the sea,' as in 'marine biology,' the study of life in the sea. Maritime is an adjective meaning 'of or pertaining to objects or actions related to or conducted on the sea,' as in 'maritime history,' which is the study of events that have occured on the high seas -- shipping, commerce, discovery. Thus, 'marine' directly relates to the sea, while 'maritime' relates to things associated with the sea.
When talking about the soldier called a Marine, he is a soldier trained to travel on ships and fight at sea. They are also used as a rapid strike force, trained to board ships and disembark on shore.
Technically, 'marine' when used in a scientific setting pertains to the salt water oceans; while 'aquatic' pertains to fresh water lakes and rivers. However, that distinction is often blurred or ignored in common speech and writing.
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