I was a big fan of the original until the Next Generation came out. In comparison, the original's storylines are more in your face and the sets and acting more campy.
Like Farkel said, Roddenberry told stories that reflected many of the situations happening back then dealing with civil rights, the cold war, etc. Kirk was a man of action and it never failed that within three minutes of the beginning of the show, he was leading a search team on the planet's surface with phasers on stun.
Times are a bit more complex now. Moral stories aren't as black and white as they were in 1968. Answers to the problems people had back then have produced other/new problems. So, because of that, NG's stories were / are a lot more philosophical, deeper, since they benefit from the passage of time since Roddenberry addressed some of the issues the first time around.
Picard, too, relfects that new problem solving technique. Rather than resort to force right off the bat like Kirk, he was more likely to think things thru... see all the angles, and use force as the very last alternative. IMO, he was a better captain.
For their respective times, they were both great series. Way ahead of their time. When it first aired, the latest NG episode was always the topic of conversation the next day at work. Sometimes we'd even call each other right after the show went off. Must see tv.