Not a second season just a second cour of the first season
There is no difference between them, 'second cour' is just a retarded way of saying season 2.
They're different because cours are released back-to-back and always number after the first one (so the second cour will start with, e.g., episode 13).
The only exceptions I'm aware of are circumstances due to COVID-19 (e.g., Re:ZERO S2), but the cours were initially planned to be back-to-back. Seasons are usually split up between multiple years.
Alright that's a lie. As @Impognagrift said, cours are in the same production cycle, and I guess that's the main difference.
A second cour is part of the same production cycle of the first one. That usually implies same studio, same staff, same production committee. A different season doesn't have the same implication. A second cour is something that's already planned before the anime even starts being made, a second season isn't.