Yes on Juliana's over Grimaldi's.
Just for background, if case anyone cares. ... since Az refused to go into it.
There was Grimaldi's. ... opened late 80s, or around 1990, I think. ... before the park went in by the water and DUMBO turned into NY's version of a Disney theme park over the last decade. The pizza was awesome.
In the late 90s, Patsy Grimaldi decided to retire, sold the name and got out. The pizza stayed pretty awesome, even though there were Russians making the pies more often than not. The coal-fired oven was the key.
Then, about 8 or 9 years ago, the new owners lost their lease and decided to move a few doors down, to a bigger corner place. They got permission to install a new coal fired oven (they are illegal now). At the same time, they decided to start franchising the name.
It's been a bit downhill since then.
A year after Grimald's moved, Patsy Grimaldi, or his family, decided to reopen in the old spot, using the old coal oven. He couldn't use his name -- he had sold it -- so in came Juliana's. The owner's of Grimaldi's tried to block him, and there was a bit of a legal battle. They lost.
For my money, I'll take a Juliana's pie any day, any time over what Grimaldi's is now slinging. Juliana's is still one of the best pies in NY. People just know the Grimaldi's name, though, and you'll find lines at both, even though Juliana's is superior in my opinion.
Regardless, there is a pizza place on Water Street called Ignazio's, across from the River Cafe, which ingeniously piggybacks off of them. They put a sign out front that says something like "the pizza," and confused tourists go in there instead, or if they aren't confused they see the lines for Juliana's and Grimaldi's and decide it isn't worth it. Just avoid that place.
I'll second most of Az's list. ... and this is something I am really passionate about.
Juliana's and Grimaldi's are sit-down, by-the-pie places. If you want the quintessential NY slice, a lot of places that would have been at the top of the list have closed down, unfortunately, but Joe's (first one on Az's list and that Moddy did a link to) is a NY treasure. Perfectly balanced sauce, good mozzerella and just the perfect thin crust that stays firm. We were actually talking yesterday about getting some pizza there this weekend. I am amazed at how they do it, since they essentially are just running pie after pie into the oven and selling 2, 3 or 4 slices at a time to people rolling through there like it is an assembly line. You'd think it would make for subpar pizza, but it's REALLY good.
The beauty of Joe's, too, is that it's right in the heart of the West Village, so there are a lot of things for you do to there, and if the weather is nice enough, you can grab a couple of slices and walk over to Washington Square Park and people watch while you chow.