Continuity mistake: During the musical montage when Neo sees again the birds fly in the same pattern, there's a bit when "Tiffany" gets her order from the counter and exits the coffee shop. Neo is eating from his poke bowl, chopsticks in his left hand. His right hand is flat on the table, but in the next shot we see him through the window, and his right hand is turned 90°. (00:26:25)
Other mistake: Neo and Trinity are pushed by a horde of zombie-like bots inside a "kill-zone." A helicopter fires rockets at them. In the wide angle of the cars blowing up, the second pair of cars explodes without any rocket CGI effect added. It's like they just blow up spontaneously. (02:06:25)
Plot hole: Considering how important it is to the Analyst that Neo stay connected to the Matrix at all times, it is unfathomable that his body in the real world is unguarded. While keeping an eye on him inside the Matrix at all times is difficult, there is no reason whatsoever that he should be left alone inside his pod with no security for even a second. This is an entirely different scenario than the original 3 films where resistance fighters finding Neo was integral to the machine's plans.
Plot hole: It's supposed to be really difficult to get even a single member of the crew into the tower where Trinity was being held, we were shown the difficulty that program Morpheus had. Then when he gets there, all of a sudden Bugs is swooped in. No explanation, made the previous 5 minutes seem ridiculously pointless.