There are two main differences between the Playstation and Dreamcast versions of the game.
The Playstation version contains a "Super Story Mode" which lets you play through the story of the manga using various characters. The Dreamcast version does not contain this option.
The Dreamcast version contains both Jojo arcade games (Jojo's Venture and Jojo's Bizarre Adventure), and you can choose to play either one. Jojo's Venture has less characters than Jojo's Bizarre Adventure, and Jojo's Bizarre Adventure has a special Challenge mode that lets you unlock playable characters when you complete it with certain characters. The Playstation version of the game is a mixture of both arcade games.
The Playstation version of the game contained a "Super Story Mode", which let you play through the game as various characters, and depending on how well you did, unlocked bonuses.
The Dreamcast version states on the back cover that the same mode is available, yet it's not. It is unknown if the error was done on purpose or not.
Wondering where most of the names of the characters are derived from? Why is there a character named Devo, as in the band Devo? Or Mariah, who might be named for singer Mariah Carey? Yep...most of the characters in the JoJo's Bizarre Adventure series have their names in the roots of many famous (and not-so-famous!) musicians! Here's just a few:
Iggy, the dog: named for singer Iggy Pop
Devo, the Devil: named for 80s band Devo
Mariah: named for singer Mariah Carey
Midler: named for singer Bette Midler
Dio: named for an obscure band called Dio; the name also means "God" in Italian
Vanilla Ice: yes, like the rapper
The Super Story Mode allows you to play through a game version of Volumes 13 to 28 of the original manga series.
Two anime series of JoJo's Bizarre Adventure were produced in Japan...the second series is a prequel series, and if you play the Super Story Mode and then watch the prequel anime, you'll notice a LOT of similarities in how the story plays out between the two (since both use the manga series as their base).