Basically, they're two different shows that cater to two different tastes.
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Star Wars: Clone Wars (2003) - The 2D animated one, is a far more moody, big on visuals light on story and dialogue "performance piece" if you will. Considering the first Volume (the first 2 seasons) of this show consists of 3 MINUTE episodes, you can blow through the entire series in about half an hour.
The 3rd season (Second Volume) consists of 5 10-20 minute episodes which gets into more story while retaining the gorgeous visual style of the previous eps. But you should also keep in the back of your mind it was conceived as a hasty intro to Revenge of the Sith (after Lucas realised he had another clusterfuck on his hands) so much of the story is devoted to the battle of Coruscant (Although to be honest the Battle as shown in this series is better than what we got in the film).
As it stands "officially", It's almost non-existant from an official LucasFilm PoV, maybe because it was better and tonally different to the films? Maybe Lucas had a bad trip, I don't know, but generally it has little bearing on THE Clone Wars, but seeing as you can get through the entire thing in about an hour, I guess its worth watching just so you pick up all the little bits and pieces.
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Star Wars: The Clone Wars (2008) is a 3D CG animated show that you see all over the place nowadays. While it shares some of the visual style of the 2D show, the colors are far more muted, and the vehicles/space battles/most of the environments are modeled "real", like they would appear in the movies. It's very much a story of the week kind of show, but one that has overreaching themes that span the entire series. Like others have pointed out, its not in chronological order (The opening 2 episodes of season 3 are a prequel, and sequel to the 1st, and 5th episodes of season 1). However, like Season 4 of Star Trek: Enterprise the series relies on 3 episode arcs to tell one big story, and these are usually always one after another. Considering each episode is a 20 minute show, it will take some time to get caught up. Luckily the series don't start again till October I think, so you have time.
So, overall, i'd say its best to watch the 2D animated show first (Star Wars: Clone Wars) purely because you can do it in about an hour, and it should set up any little bits and pieces which might go unexplained in the CG show (Star Wars: The Clone Wars).