The article states that the "first remontoire was invented in 1595 by a Swiss clockmaker named Jost Bürgi." It goes on to explain how this mechanism relies on a spring. Immediately afterwards, the article says that it "wasn’t until 1739 that the spring remontoire was invented by the famous English clockmaker John Harrison".

Could someone explain to me (as a layman as far as watches are concerned) what Harrison contributed to be honoured with inventing it for a second time? Or is there a technical difference between a remontoire with a spring and the term spring remontoire?

So it's kind of like a mechanical decoupling capacitor?

The remontoire has a technical/practical modern use in the Zeitwerk.