In modern day, there is no difference. Captain is the most common. In the oooooold days (hundreds of years ago, say, 17th-18th century), “Master and Commander” was a specific rank. Usually people who were captains of specific types of ships were actually granted this rank. In the late 18th century, it got simplified to just “Commander”.
In the modern Navy (US and I believe Royal as well), “Commander” is rank O-5, one lower than “Captain” (O-6). These are ranks, but a Commander can still be a “captain” by virtue of having command of a ship.
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