Gogeta and Vegito are the fusion of Goku and Vegeta through different means. They might have different power levels and fusion time, but do they have the same personality or are they basically the same?
Most People seem to think Gogeta is the serious counterpart of the fusion and takes after Goku while Vegito is the cocky and confident counterpart and takes after Vegeta. This is incorrect. The main reason people seem to think this is because of the Gogeta in Dragon Ball Fusions: Reborn who is completely different from the canon iteration of Gogeta.
Firstly, all fusions share 1 common trait. They tend to become very cocky and act in a very arrogant manner while mocking their enemies. This personality seems common between Vegito, Gogeta, Kefla, Gotenks etc.
Gogeta seems to have the same playful and cocky personality Vegito does. We know this because, after fusing, despite the brevity of the situation, they spend time thinking of a name. Even when he starts fighting Broly, Gogeta was smirking and smiling throughout the fight and was never really keen on putting an end to the fight right at the start(Even Though he could) and was simply using more power than his opponent and overwhelming him.
However, another important thing to note is that Broly is somewhat of an anti-villian and he wasn't really someone you could communicate with while fighting. So Gogeta didn't really have the necessity to insult him or mock him.
Not to forget, although SSJ4 Gogeta isn't canon to the series, he was probably the complete definition of a troll when it comes to all fusions. The Bluff Kamehameha & Back Scratch etc.
In conclusion, there don't seem to be any unique defining traits that distinguish Gogeta and Vegito apart from their appearance. So no, their personalities seem to be the same.