As everyone knows, several spin-off series were created a generation later. The TNG pilot aired when I was an undergrad at UCI. Gene Roddenberry was still helping in production so the original character concepts, and several of the story lines, were rather good. It took some time for the cast to settle in and a lot of the scripts were quite preachy - taking today's social issues and transferring them to alien worlds. This was, of course, an aspect of the original series as well, but not nearly as obvious or cliche. Then again, maybe I noticed it more because TNG's issues were in my own time, not the issues of an earlier one. But the characters, especially Picard and Data (both played by wonderful actors), came into their own and there was enough good material to hold it up.
I can't say the same for Voyager and DS9, whose plots became increasingly copy-cat of other material or just plain ridiculous. These shows were created without Roddenberry's influence, as he had passed away shortly into the airing of TNG. You can even see the PC influence seeping into the casting - let's have the first female captain (Voyager) and how about the first African American captain (DS9)! Avery Brooks' performance as Sisko was so horrible I have to assume their choice was motiviated by other factors. Then you have Chakotay as an obvious alien version of a Native Amercian. The original series always did have an international cast, but that fact was secondary to the characters themselves. That aside, I think Kate Mulgrew made a decent captain in the end and adding Jeri Ryan as a recovering Borg helped to introduce the same character dynamics created by other "no-emotion" characters like Spock and Data.
But at the end of the day, I still maintain that nothing will ever beat the original classics. No matter how bad the special effects by today's standards, or even the acting, the stories and the concepts are up there with the best, and that is where the heart of sci-fi really lives.