Deadpool is tough, funny, and charming, but let's face it: he's not exactly a looker. The experimental surgery that gave Wade Wilson his regenerative powers also kicked his cancer into overdrive, and while the procedure kept tumors from ravaging his body, it didn't keep them from growing all over his face. As a result, Deadpool looks like "a testicle with teeth"—his words, not ours—and prefers not to show his face in public.
When he's on the job, that isn't much of a problem. Put a mask over that gruesome visage and nobody can tell the difference. During Wade's downtime, however, it's a different story. If he wants to go out, Deadpool needs to either suit up, deal with being a walking freak show, or find another solution.
In the comics, Deadpool chooses the third option. Thanks to a holographic image inducer—the same technology used by X-Men like Nightcrawler and Beast to enjoy a night on the town—Wade can change his appearance and wander around incognito. That's a pretty simple solution, so of course Deadpool finds a way to screw it all up. Wade's brain is just as broken as his body—to put it charitably, he's psychotic—and unless he's on a specific mission, he uses the image inducer to change his appearance willy-nilly.
Shape-shifting may not be quite as obtrusive as a face that looks like Freddy Krueger's, but it's awfully close (and those yellow dialogue bubbles give tend to give him away, anyway). Eventually, Deadpool decides to stick with the Teenage Mutant Ninja Turtles' time-proven, low-tech disguise: a fedora, a trenchcoat, and the comforting knowledge that in New York City, a tumor-infested mutant probably won't be the weirdest thing you'll see wandering the streets.