Factual error: In the second case, Phoenix arrives on the scene of the crime just 11 minutes after the murder occurred, but examining the body, he finds the murder weapon "encrusted with dried blood." In that short of a timespan, the blood would be anything but crusty - especially considering that the smell of blood is so strong that Phoenix can feel it from the corridor.
Character mistake: When the first case's true murderer is KO'd by Phoenix's amazing attorney skills, he is carried away from the courtroom. The Judge then asks to the prosecutor, "Mr Payne, your client?" He should have obviously asked, "Mr Payne, your witness?" And that's exactly what he asks in the Trilogy remake, that fixes this mistake.
Continuity mistake: During the first case, when Mr. Sawhit has the first of the series' patented witness breakdowns, he angrily tosses his toupee at Phoenix. However, in the courtroom shot from behind, he still has a full head of hair.
Other mistake: One of the slip-ups that incriminate Redd White involves the way the victim died. Phoenix points it out, saying, "The victim died from a 'single' blow!" Using quotation marks for "single." The autopsy report, though, does not literally say "single," but just that died from "a blow" without ruling others out. If this seems minor, consider that the Trilogy remake feels the need to remove the quotation marks, obviously because it was not quoting the in-game text of the autopsy report, which was much more vague.
Character mistake: During the tutorial, Phoenix needs to be reminded a few things. The judge asks him the victim's name, and Mia Fey next to him tells him where to look to get that information. Her line though is flubbed, since instead of "the victim's name" she says "Look, the defendant's name is in the Court Record." This mistake was spotted and fixed in the "Ace Attorney Trilogy" remake.