Do the Tom Clancy movies qualify? The Hunt for Red October, Patriot Games, Clear and Present Danger, etc?
Oh, and how about Kiss the Girls and Along Came a Spider?
Mister Roberts --> Ensign Pulver
The Hustler --> The Color Of Money
Mad Max --> The Road Warrior
Often called Army of Darkness: Evil Dead 3, or even Bruce Campbell Vs. The Army of Darkness.
In the Heat of the Night was followed by They Call Me Mister Tibbs and in turn by The Organization.
The Quatermass movies had relatede titles in England (Where “Quatermass 2” is reportedly the first movie to use the “2” in its title), but in the U.S. the titles of the same movies are unrelated. The Creeping Unknown followed by Enemy from Space followed by Five Million Years to Earth.
Come Back, Charleston Blue is not a sequel, as the unwary might expect, to Go Away, Charleston Blue, but rather to Cotton Comes to Harlem.
At one point they wanted to call it The Medieval Dead, but the studio wanted it to stand alone, especially since they hadn’t released either of the first two movies. 
Do we want to count *Raiders of the Lost Ark * and Indiana Jones and the Temple of Doom?
Didn’t Lucas officially change the title of the first film to Indiana Jones and The Raiders of The Lost Ark?
Sir Rhosis
First Blood was followed by Rambo.
That’s the label on the DVDs now, but it wasn’t that way originally. Also, George Lucas and his revisions can kiss my butt. 
The title was Rambo: First Blood Part II, so I don’t think that counts. There was, however, Rambo III, which I guess does count.
The only real Cheaper by the Dozen was followed by Belles on Their Toes, which were also the titles of the books upon which they were based.
Roger and Me was followed by Pets or Meat.
A fistful of Dollars,
For a Few Dollars more,
The Good, The Bad and The Ugly.
Get Shorty > Be Cool
El Mariachi => Desperado => Once Upon A Time In Mexico
The Year My Voice Broke was followed by it’s sequel Flirting.
Sinatra vehicle Tony Rome was followed by Lady in Cement.
The movies were based on the Virgil Tibbs series of books.
Truffault’s “Antoine Doinel” films: The 400 Blows, Love at 20, Stolen Kisses, Bed and Board, and Love on the Run (US titles, but the original French titles had no similarities).
That may be, but they perfectly meet the requirements iof the OP, just like the Bond films.