Think of that initial first-person bit seen in the movie's teaser trailer, which moves the camera inside Parker's head and lets you see skyscrapers of Manhattan through the eyes of Spider-Man.
Web Rush is an occasional guided, contextual mechanic, rather than a persistent way to play The Amazing Spider-Man. Instead, Web Rush lets you point toward a spot in the environment, hit a button, then watch Spidey maneuver there in acrobatic fashion.
For example, if you're on top of one building and want to spin a web to another, you line up the cursor and hit the button. Spidey then runs across the roof, swings over, runs up the building and does a neat little flip to land in the designated spot automatically. Its use slows down time, making it applicable during combat, but Beenox didn't demonstrate the feature in a more intense situation.
Outside of a freely navigable Manhattan, Web Rush is easily the biggest new feature in The Amazing Spider-Man, which picks up the story after the events of the upcoming film reboot. Swinging around Manhattan looks as you'd expect, save for a tighter camera Beenox is implementing in an attempt to increase immersion. The camera is close in on Spidey's shoulders most of the time.
Manhattan is large and populated with objects Spidey can interact with, like flagpoles. A Beenox rep showed off Times Square, but wouldn't take me for a ride through the entire city. What I saw looked the part for its New York City setting, and I suspect fans looking for a beautiful return to free-swinging won't be disappointed.
Combat in The Amazing Spider-Man shows a mixture of influences -- hand-to-hand combat is straight-up inspired by Rocksteady's Batman games, employing contextual dodging and countering mechanics for a basic, two-button combat system. Before Spidey leaps into the fray to deal with groups of armed thugs, he can silently stalk his prey and take them out, reminiscent of the Noir Spider-Man sequences in Spider-Man: Shattered Dimensions.
There's still a lot of development left before The Amazing Spider-Man launches in June, and Beenox claims it's taking steps to ensure The Amazing Spider-Man isn't hit with the same criticisms that Spider-Man: Edge of Time received. With less constrained environments and the return of free web slinging, the studio is hoping fans will be eager to become Peter Parker once again.
The Amazing Spider-Man launches on June 26.