I was in Manhattan last weekend and decided to try Lombardi's and Grimaldi's back-to-back. I ordered a plain cheese pizza at both places.
In my personal and humble opinion, Grimaldi's was the better pie. First, the crust on Lombardi's was tough, chewy and burnt in spots. The burnt spots had an overpowering charred flavor. Grimaldi's, on the other hand, had a softer crust with no strong, burnt spots. Second, Lombardi's sauce has a stronger flavor. Some may prefer their sauce, but I felt it was a little too strong. Third, the cheese on Lombardi's pie had that stiff, fake-cheese feel you find on inferior pizzas. Grimaldi's cheese was softer and gooey-er. For my money, Grimaldi's is the better of the pair.
FYI: There's a Grimaldi's outpost in Phoenix, AZ which is a miserable copy of the original. The pizza is absolutely atrocious there. Thankfully, Pizzeria Blanco serves a fantastic super-thin pizza in Phoenix. Probably the best thin pizza I've had in the US.