Some of the things in the 2003 Clone Wars was discarded, like the detail of Grievous getting his ribs crushed my Mace at the end of the 2003 series, but then in the newer Clone Wars, he already has the coughing.
According to Lucas, the coughing was intended to be a result of his cybernetics, at least in his mind when they crafted the character. It was supposed to represent how this was the prototype technology to what made Darth Vader.
The cough in the original CW series was a completely separate event. Lucas saw how ridiculous Grievous was in this series and saw how out of line it was for his view of Star Wars and had the Mace Windu scene put in to help give an explanation to the general audience what happened between the series's incarnation and the movie incarnation.
Ultimately, I think GL has used TCW as his method of going back and "fixing" (for better or for worse) concepts and ideas in the EU that he had no direct control over before. This is just my opinion, but I think the scene in CW was just a quick fix and GL has now allowed TCW to give the "correct" version of Grievous as he was always intended to be.