Beyond the 'expected' changes of choosing a different advanced class (different class skill, mobility, base stats, weapon proficiencies), each class has stat caps associated with it. You can tell when a character is at the stat cap for a respective stat of their current class when it is displayed as a green number in their character stat screen.
The reason you may want to do this is if you are trying to maximize your total stats with a character. There are some advanced classes that have a higher stats than others when you sum the caps of all their stats. Or, you may want to reclass to one that has high stat caps for stats you want to prioritize.
You can see what stat caps apply to each class here (be forewarned that there are some classes masked here, because they may apply to only certain characters or are involved with the later stages of the story):
https://serenesforest.net/engage/classes/maximum-stats/