Both are good and useful phrasal verbs and can be use with a degree of variety and overlap. I would use move forward more in a situation where we have a plan or agenda. For example, in a meeting or project. Move on, I would use more for progress through life. I want to move on after my break-up.

Hello there, I'm not a native speaker nor teacher but below is my understanding, hope it helps :) Move forward: when you want to talk about heading to a more positive situation from something, from somewhere to somewhere. Ex: Life is too short for us to keep looking back, we need to move forward in order to seize the moment. Whereas "Move on" is more sentimental & usually is used when people want to talk about relationships. Ex: After a few months of suffering from her divorce, she was ready to move on.

It helps a lot. Thanks.

Yes actually "move on" is more common where as both have a similar meaning but "move forward" literally means the direction you are going and move on means to leave your current situation and find a new one usually talking about a relationship.