we all wear masks on our faces and on our hearts
Godot: Of course, being extradited from Hell is a tedious affair. The meticulous regeneration and adjustment of all your internal organs is... Well... Let's just say modern medicine allows us all to live to a ripe old age. Even someone like me.
Phoenix: So... you mean... That mask you wear is...?
Godot: This ugly device? I promise it's not a fashion statement, my unenlightened friend. Without this, I can't see your frequently dumbfounded face.
Godot's mask is his most distinguishing feature, but what it symbolizes is much deeper than that: his transformation from Diego Armando to Godot. The mask of the man "Godot" hides him from the world and allows him to be an enigma, without the trouble of any background to get in his way — not even his coworkers in the police department know who he is (by which I mean Gumshoe, the face, eyes, and ears of the department, albeit not brains).
The mask hides more than just his identity; it also hides his heart, as he says himself. Godot clearly exhibits disdain and scorn for Phoenix, but does not reveal why until the entire truth comes out in the last case. Up until then, he is carefully guarded and hides the truth behind biting sarcasm and mysterious philosophies about coffee. Most importantly, what the mask hides is his own shortcomings: how he ran from Mia's death and tried to escape the reality of it. Metaphorically speaking, the mask is his pride. While he wears it, he can turn his back on what happened and focus entirely on getting his revenge on Phoenix Wright. He knows, deep down, that he's just using it as an escape, but he's too proud to admit he failed.
The truth is, the only person I could never find it in my heart to forgive... was me. I was the one that failed to protect Mia... Me and no one else. I tried to avert my eyes from the truth... to escape from the harshness of reality. I just couldn't face Mia's death head on, so I ran!
You can't really blame him. After all, the man lost five years of his life, and the one person he hoped would be waiting for him wasn't there when he woke up. He had to deal with that somehow, and the method he picked was to run away from it all and hide behind a mask. If Diego was in therapy, his therapist would probably tell him to stop ignoring his true feelings and face them. (Then again, this is a video game character, and video game characters do not have logical, real-world things like therapists, no matter how many of them could really benefit from one.) The point is, it's a human thing to do, and gives him a relatable motive for his actions.