When Joffrey becomes king, he seems to give equal weight to both the Lannister and Baratheon sigils. Why is this? Robb Stark doesn't consider himself a Tully. It's almost as if Joffrey himself knows he's not a Baratheon.
Three people were flying the stag. Stannis changed his as well. Not sure how Renly the guy with the weakest claim wound up with the old school stag though. The lion was probably added as a PR move to remind people where the most powerful house stood.
Joffrey considers himself a Lannister, as much so, if not more than a Baratheon. This is probably due in large part to Cersei's influence, and Robert's seeming indifference towards him. It is commented on as unusual by several characters throughout the books.
It's actually common heraldic practice and it would be called marshalled arms. The simplest is a division per pale, then inserting the different charges on either side, as Joffrey's is divided with the lion and stag. This is usually done to show a union of two groups or to join your arms to a symbol of office. It could be either in this case, as Joffrey's a product of both houses, or
he's not really a Baratheon, and he's joining his true arms to those of the current royal family which has been functioning as the royal device to further his claim.
While there may be deeper symbolism behind it, I expect the fact that is traditional heraldic practice to be the main cause. While reading the books, I have noticed Martin paid particular attention to description of devices and followed other common themes such as canting. This is basically putting puns in your device, such as house Waxley having candles, or Heigh (which sounds like 'hay') having a pitchfork.
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 has also been relegated to being a cadet house, and continues to fly the traditional banner and sigil of House Baratheon of Storm's End.
Thus the reason why Joffrey changes his banner and sigil is to distinguish his house, House Baratheon of King's Landing, which represents the union of House Lannister and House Baratheon, from the cadet Baratheon houses.
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.
Expanding my comment: Splitted coats of arms usually created by inheritance of a realm by maternal side, which traditionally by heraldics placed on the right side of the shield next to the paternal side. For Joffrey if fulfills this rule, the Stag goes to the left side facing right, and the lion goes to the right side facing left. Also express the claim the rightful rule on the Lannister house, unlike his father who had nothing to do with Lannister realm, since he had a wife from there without direct right of rule over it.
Furthermore it is clear that Joeffry is the next in succession in the Lannister house, since his uncle (Jaime) is in the order and Tyrion won't inherit Lannister's realms because of his conflict with his father. Therefore he rightfully has claim on Lannister realms as well as Baratheon realms.
Joffrey, although a Baratheon heir by lineage, does not enjoy any actual, meaningful support from the Stormlands. This is split between Renly and Stannis intially. Beyond birth names and symbolism, the true dynamics of power in A Song of Ice and Fire are that Joffrey is a Lannister king put and kept on the throne by Lannister wealth and resources. He has no true Baratheon support. And that is why the Lion is as prominent as the Stag for Joffrey.