In the original series, you had Vinny as the comedic character, but most of the other characters were more serious. In this series, you have her father, her younger, brother, and her best friend as more comedic relief characters. For the modern series, they leaned more towards the comedy, which makes sense because it's Disney and they are aiming at tweens/teens. The older series was more of a dramedy aimed at slightly more adult audience.
If you haven't seen it, the episode of the original series on the 1992 L.A riot is a fantastic episode.