Rate is the paygrade of an enlisted sailor,
rank is the paygrade of an officer (In the US Navy anyway).
Rate titles are
For paygrades E-1 E-2 E-3: Seaman (Fireman, Airman) Recruit, Seaman (Fireman, Airman) Apprentice, Seaman (Fireman, Airman)
For paygrades E-4 E-5 E-6: Petty Officer 3rd Class, Petty Officer 2nd Class, Petty Officer 1st Class
For paygrades E-7 E-8 E-9: Chief Petty Officer, Senior Chief Petty Officer, Master Chief Petty Officer
Rating is the job an enlisted sailor does. Examples include Electrician's Mate, Boatswain's Mate, etc.
In practice, the two are usually combined to give something like Electrician's Mate Third Class Smith. This would be Smith's specific rate (as opposed to a generic Petty Officer Third Class Smith.).
Edit to add: A more in-depth explanation:
https://www.usni.org/magazines/nava...il/bluejackets-manual-ranks-rates-and-ratings
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The dialogue doesn't come across that way:
"Seaman Jones here is into music in a big way, and he views this whole boat as his own personal, private stereo set."
Chief Watson isn't framing the story like "Back when Petty Officer Jones was just a seaman," he talks about him using what is meant to be his current (but incorrect) rank.
It does to me because it is set up by the COB telling Seaman Beaumont:
"If he gets to ragging on you too bad, ask him about Pavarotti."
to which Jonesy replies with
"We don't have time for sea stories."
So it's a past event and given the exchange about the 'one way out at Pearl' a story the COB has told numerous times before.