The reason why is because Joffrey is of House Baratheon of King's Landing.
Stannis flies the banner of House Baratheon of Dragonstone because Robert installed him there and made him Lord of Dragonstone, thus creating a cadet branch of House Baratheon. His banner and sigil is a stag surrounded by a heart of fire, appropriate for a follower of R'hllor.
Renly thus becomes the Lord of Storm's End, which while it used to be the main house, ishas also been relegated to being a cadet house, butand continues to fly the traditional banner and sigil of House Baratheon of Storm's End.
Thus the reason why Joffrey changes his banner is because itand sigil is the banner ofto distinguish his house, House Baratheon of King's Landing, which represents the union of House Lannister and House Baratheon, from the cadet Baratheon houses. It is
Robert most likely maintained his old sigil as he felt he was from Storm's End. However, given Joffrey's pride and arrogance, it would make sense for him to adopt a new sigil and banner for himself, as the first heir of House Baratheon of King's Landing to have been born in King's landing. He views himself as the culmination of both house's, evidenced by his mockery of Tywin's "hiding under Casterly Rock" and his general disdain for Renly and Stannis.
During the War of the Five Kings, this new banner and sigil served to both distinguish himself from the other claimants of the Throne, Renly and Stannis, as well as to remind the kingdom of his Lannister ties.