In general, ST:TNG was better done. It had higher apparent production values and the characters were better-developed. This does not make it the better show.
ST was more exciting. Kirk and company were usually saving planets, eliminating space monsters, or overthrowing repressive governments, Prime Directive be damned.
The crew on ST got out more. Nearly every episode involved a trip to an alien planet. Granted, these were often sets and looked like it. ST:TNG has many more episodes taking place entirely on the Enterprise. This gives the show a more claustrophobic feel.
There was too much techspeak on ST:TNG. Writer Ronald Moore takes about it here.
Moore said he used to put the word tech in his scripts as a placeholder, which led to stultifying dialogue like this:
Picard: "Mr. La Forge, I need you to tech the tech."La Forge: "But Captain, if we tech the tech then the tech will override! The tech main engines might tech too much!"
It didn't seem so bad at the time but looking back at the show, this is a major fault. It also shows up with the number of "level 1 scans" and "level 2 diagnostics". They tossed these terms around constantly like they meant something.
ST:TNG was closer to real military structure. On Star Trek, Kirk was in charge and Spock was second in command. After that, the chain of command became murky. Any time Spock was in charge, one or more crew members became insubordinate. Then there is the whole issue of the bridge crew being the first to beam into a hostile situation.
ST was brutal. Having characters die, even red-shirted extras, was a new thing in the late-1960s. This was cutting-edge at the time. I remember watching the show in first run and being shocked a few times when someone died unexpectedly.
The crew members of ST:TNG were too competent. Anyone could make anything from anything. Warf could generate a force field from a communicator. Data could make anything from his own spare components.
The Holodeck was too real. People were constantly getting trapped in the holodeck. In one episode, Picard and Data couldn't even tell that they were in the holodeck. It was just a room. You should be able to walk in a straight line until you hit a wall then feel for the door.
ST only had two good seasons. When it was in first-run, I knew that the third season was the last. The show felt tired. Most of the episodes revolved around Kirk falling in love (again).
The first season of ST:TNG stank. Episodes were just plain boring. Some were also dumb like the time Wesley was sentenced to death for breaking a window on the planet of joggers (seriously). The first season seemed bad at the time and it has not improved with age. In contrast, some of the second season episodes are among the best.
Even though ST:TNG lasted longer, it did not run out of energy. Some of the episodes in the last couple of seasons were weird but the show never felt as tired as the original did in its last season.
Both shows were much better than Deep Space 9 or Voyager.