At one time the main difference was in the ratings and accessories.
Typically the same set will be rated 12% higher for standby than for prime power.
The prime power set may have enhancements such as a larger oil sump or an oil cooler or possibly a larger radiator. (Though many standby sets also have oil coolers.)
A standby set gets old with very few running hours.
Standby sets are pushed to the maximum output.
Prime power sets are typically derated so that after 10,000 or 20,000 hours, they will still be capable of delivering rated output.
Admittedly my information is dated as I started to get heavily involved with standby sets about 20 years ago.
I did a lot of research on gen-sets.
I compared engine model numbers and generator end model numbers between standby and prime power sets.
The same basic engine and generator end would be used for a standby set at one KVA rating with a Catalogue number roughly indicating the KVA capacity and on another prime power set with a different Catalogue number and capacity of 25% less.
Eg: standby rating, 100 KVA, Prime power rating of the same basic set, 80 KVA.
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Ohm's law
Not just a good idea;
It's the LAW!