Spin-offs are common in Television and continuity isn't one of them. When a character is popular off one show, they are given their own show, a spin-off. In this post, I discuss how the character might have changed in their new show than in their original incarnation.
Cleveland Brown in "The Cleveland Show"
Original show: Family Guy
He started out as Peter Griffin's slow talking Deli Owner friend and neighbor. When the show was canceled and returned, his personality became more vibrant. Most fans are surprised to find out the actor (Mike Henry) that does his voice is White and not black. Before he gained his own show in 2009, his personality was slightly getting more brash but on the show itself he is more outgoing, dropped the slow voice and he is much more sexual and violent. Various flashbacks give conflicting histories of his speech patterns. His show was canceled in 2013 and he has since returned to "Family Guy" in which he is not the main focus but has been more or less the same but slightly reduced.
Rhoda Morgenstern in "Rhoda"
Original Show: Mary Tyler Moore Show
Rhoda was Mary's friend and neighbor on MTMS and then she got her spin-off. She starts out single, then meets the character Joe and they get married, but then ratings dropped. Producers thought fans didn't like to see her happy. In season 3, they decided to have them divorce which split fans apart and made them much more mad. Ratings dropped even further in seeing Rhoda unhappy. Upon moving to her own show, the mention of Rhoda’s Jewish religious and ethnic background faded. While on the parent sitcom, Rhodawas explicitly Jewish. During the first season of Rhoda, the representation of Rhoda Morgenstern altered from her parent show to fit a more mainstream audience: she was trimmer, more confident, and less "Jewish."
Captain Jack Harkness in "Torchwood"
Original Show: Doctor Who
The character's introduction served to posit him as a secondary hero and a rival to the series protagonist, the Doctor, simultaneously paralleling the Doctor's detached alien nature with Jack's humanity and "heart". Jack's original appearances in Doctor Who were conceived with the intention of forming a character arc in which Jack is transformed from a coward to a hero, and John Barrowman consciously minded this in his portrayal of the character. In Torchwood, Jack is re-introduced as a changed man, reluctantly immortal, having spent years on Earth waiting to reunite with the Doctor. Jack returns with a lighter attitude in the second series. Within Doctor Who, Jack's personality is relatively light-hearted, although this changes in Torchwood's first series, where he becomes a darker character.
Joey Tribianni in "Joey"
Original Show: Friends
"On Friends, Joey was a womanizer, but we enjoyed his exploits. He was a solid friend, a guy you knew you could count on. Joey was deconstructed to be a guy who couldn't get a job, couldn't ask a girl out. He became a pathetic, mopey character. I felt he was moving in the wrong direction, but I was not heard." —Kevin S. Bright on the reason for Joey 's cancellation.
Sam Puckett and Cat Valentine in "Sam & Cat"
Original Show: Cat from "Victorious" and Sam from "iCarly"
In iCarly, Sam was a wild card, a criminal, morally ambiguous, angry and crass. She wasn't the man character obviously. In "Victorious," Cat was flightily, dreamy, soft voiced and loony. I was against the idea of pairing these two. Because in the original show "The Odd Couple," Felix is straight and neat while Oscar thinks out of the box and is sloppy. You need your 'straightman.' In this case, neither of these characters were the 'straightwoman.' "iCarly" had ended as the main actress wanted to move on and "Victorious" was forced to end in order to pull off the spin-off. Both are wild cards. Sam is now a biker-chick but still criminally-minded and crass. Cat is the same, soft-voiced and flightily. But somehow they tried to wedge Sam into the straightwoman position, reigning in the insanity of Cat but it did not quite gel. The show was popular but it was not canceled because of that but conflicts behind the scenes with the actresses.