Both terms have a specific legal meaning. One requires assistance, the other does not.
Nope they have not a legal meaning.
Both terms do not exist!.
The terms you are looking for is "Mayday, Mayday, Mayday" (the real distress 'distress call')
And "Pan-Pan, Pan-Pan, Pan-Pan" ('request for assistance', the 'other distress call' for technical or medical problems)
And you most certainly HAVE to assist in a "request for assistance" call, which is Pan-Pan, Pan-Pan, Pan-Pan when a person is in life threating condition. Americans of course suggest in such a situation to call a Mayday. Official procedure for a medical condition, regardless if life threatening or not, is Pan-Pan, not Mayday.
Why is that so? So that Coast Guard or On Site Operators an distinguish between Mayday and Pan-Pan, as you can have both same time close together.
And on top of that: Pan-Pan and Mayday: are both distress calls. One means: I need assistance, and one means: we are about to lose the ship and all hands on board are in danger.
There is no "requires assistance" level of urgency in maritime radio traffic. Comments like that are by amateurs who have no idea about what they are talking.
All three (or four, depending how you count) are distress calls. In case you wonder:
1) Mayday, Mayday, Mayday: [my ship, position etc.] have problem: fire, sinking etc. Pirate attack
2) Mayday, Mayday, Mayday: [my ship, position etc.] I OBSERVE [report about what you observe]. E.g. a burning ship. A person drifting in the water, etc. A Mayday call on behalf of someone else in danger, who can not call himself.
3) Pan-Pan, Pan-Pan, Pan-Pan: [my ship, position etc.] I have trouble ... E.g. broken mast, out of fuel etc. Or medical emergency.
4) Securite, Securite, Securite: [target audience], problem, [affected position, probably your own, TIME]. E.g. Carrier with flight operations, lost cargo drifting in the water, submarine about to surface, lost ammunition (e.g. mines about to be exploded or disarmed), life fire exercise, or any Mayday operation in progress.
5) Silence, Silence, Silence - MAYDAY.
First three are distress calls.
Number 4) you have to argue, it is basically a warning.
Number 3) makes you obliged to help if it is a medical emergency.
Number 5) Calling for people who are not involved in the currently running rescue operation to keep radio silence on channels 16 and 9, and maritim helicopter operations channel 6.
P.S. From the summary: radioed Mark Zuckerberg's yacht -- which was closer -- and they repeatedly refused to respond.
That is not the same as "refused to help" - they genuinely might have not received it (which is of course not plausible, and would be super bad seamanship). Completely plausible is, that the calling boat had set his radio power to low output and the other boat was out of range to hear it. The coast guard would hear it, despite being farer away, as they have much more sensitive equipment on much higher ground. So if the coast guard "over heard" a 'conversation' between the boat in distress and Zuck's yacht, it is still possible that Zuck's indeed did not hear the call.
P.P.S. the keywords Mayday, Securite, Silence are pronounced French.