Revealing mistake: During the fight scene between Neo and the agent in the subway station, after Neo incapacitates the agent and flips out of the way of the oncoming train, you can see where the wire harness connected to his back. You can also see one of the wires visible by his wrist.
Continuity mistake: The first time Neo comes out of the Loading Program, and is about to throw up, it shows you a shot from his eyes and the floor is clear of anything, and then it does a shot of him throwing up and he does so on a bunch of wires. The angle changes too - the POV shows all the lines in the grate at about 45 degrees, but when he throws up, they're all horizontal in relation to him.
Other mistake: When Trinity confronts the agent on the rooftop and says, "Dodge this", she points her gun to temple of the agent while his head is turned to her. He glances back, but never fully turns his head. The next scene shows him being shot facing Trinity...with a shot in middle of his forehead.
Continuity mistake: In the beginning of the movie when Neo gets the FedEx envelope, he is shown at his desk opening the envelope and then pouring it out. As he is pouring the phone out, with the green packing label on the top, the shot cuts "seamlessly" to a close up of the phone being poured out the rest of the way. However, the envelope is now flipped over. The FedEx logo is now on top (the packing label is facing down).