I've read quite a bit of these books due to the Dynamic Duo archives, the two Showcase Presents volumes, and my own back issue collection. Some thoughts:
The initial change from Sheldon Moldoff doing a Bob Kane impersonation to Carmine Infantino and Moldoff being inked in a manner closer to Infantino, was a breath of fresh air. As an action storyteller, Infantino was at the height of his abilities and a lot of the fight scenes just pop off the page. The new Batmobile worked well too and the whole design sense caught back up with the rest of the industry. And, Infantino was the master of the story hook cover. This is one of the best eras for covers.
Story-wise, if they didn't go all the way back to Batman's original dark roots, they at least went back to the more straightforward late 40s era and ditched all the bad ideas of the 1950s. And although I wouldn't call the books sophisticated, they weren't as simple and childish as it was before and was basically of a piece with the rest of DC's superheroes. That's an era when DC first started introducing running subplots and The Outsider played a prominent role in the books. To a lesser degree Zatanna's quest also fit in.
To me there are several eras of the "New Look" and it's a mistake to lump them all together. There's the early era which emphasized crime fighting and detective work but didn't feature much for costumed criminals. "Castle of Wall to Wall Danger" and "Two-Way Death Trap" being two very good examples of the strength of that approach. And after an initial period of experimentation, a return to costumed bad guys but with a twist or two. "The Remarkable Ruse of the Riddler", the Getaway Genius, the Cluemaster, and, perhaps especially, "Death Knocks Three Times" are strong stories and the feature really had found its footing.
That was followed by an emphasis on emulating the television show. There's some good stories here, "The Million Dollar Debut of Batgirl" springs to mind, but my reading I get the sense that editorial is being pulled in two directions, keep up with the rest of the industry or cash in on the tv show, and I don't think either direction was entirely satisfied.
After the end of the show, I think things returned to a more natural progression. Things turned more serious again, Bob Kane basically retired and Gil Kane, Bob Brown, and Irv Novick came on board as artists, and a more down to earth approach, and Frank Robbins as writer with an emphasis on detective work. There also was an emphasis on Robin starting to grow up and the 60s social scene came into play. Robin leaving for college wasn't some out of thin air development. Some good stories, "Die Little, Die Big" for instance, and the seeds were laid for what would be the 70s Batman starting with "One Bullet Too Many" and "Secret of the Waiting Graves".