The new show, in my opinion, is actually just like G&F, as far as the storytelling goes. So the answer of which show is better depends on whether you're wearing your nostalgia glasses.
Garfield and Friends is still the best I think... it was my favorite cartoon back in the day and it's still surprisingly awesome today. The Garfield Show is OK, it certainly doesn't suck, but I'm biased towards Garfield and Friends!