Springfield Elementary School is yellow in some episodes and
white in others, sometimes changing color multiple times in the same episode (later episodes corrected this continuity problem by making the school's official color yellow, and including a picture of the "old, white" school in Principal Skinner's office.)
When wearing her normal green dress, Marge wears a red necklace. However, often she is shown for a single frame or two wearing a white one (generally this happens in close-up shots of her head.) This happens in numerous episodes.
Waylon Smithers was designed to be "Mister Burns' white sycophant," as it "would be a bad idea to have a black subservient character" putting up with the abuse that Burns doles out. (Quotes are from David Silverman, one of the show's main producers.) Ironically, Smithers is colored as an African-American in his first appearance The Simpsons: Homer's Odyssey, due to color stylist Gyorgyi Peluce's misinterpretation of the new character. The mistake was quickly caught and corrected starting with the immediate next episode The Simpsons: There's No Disgrace Like Home.
"In one episode they say this, but then in another episode they say that, and in yet another episode they say the other." As this is an animated comedy series, the emphasis is clearly on laughs rather than complete verisimilitude. Efforts are certainly made to create a vaguely consistent setting in which mostly consistent characters live and work, and many episodes make reference to older episodes, but rigid consistency of every single detail in all episodes is unnecessary. Sometimes the fact that the episodes are inconsistent is the point of the gag and acknowledged by moments such as Homer saying "Didn't this use to be like that?" or Marge breaking the fourth wall to narrate "You may remember that things used to be different..." Each episode is expected to be consistent within itself, but inconsistencies between different episodes will not be listed here unless they noteworthy for another reason. E.g., inconsistencies repeated in multiple episodes (such as the hair and skin colors of secondary characters, and the layouts of the main landmarks), drastic changes to a character's nature (such as Ralphie's school status, Jasper's abilities, or Milhouse's hair color), or something with an interesting anecdote behind it (such as Smithers' skin color).
Fat Tony's real name changes from episode to episode, but this is probably intentional because it is common for Mafia hit men (in popular culture, and possibly in reality as well) to have many different identities.