Differentiation of self can be summarized as the time available to an individual between an emotional response and their reaction. They react based on the options they believe are available to them and their personal objective regarding the situation.
To understand differentiation, imagine a person looking for a parking spot at the grocery store and its packed. This person has guests arriving for a dinner party in 45 minutes, and they are missing a key ingredient. They drive around (and around) looking for a spot, and they find one where the current car is backing out. Once it is gone, they will pull in, but then from the other direction, another car swoops in and steals the spot, ignoring that they were their first.
How would this person respond? How would you respond? Most people are going to feel angry on some level. We can use words like enraged, frustrated, annoyed, irked to indicate the level of anger. But regardless of the word we use, feeling some level of anger would make sense. Our reaction to the feeling of anger would indicate our level of differentiation.
Suppose a person has a lower level of differentiation. In that case, there will be minimal time-elapsed between feeling angry and taking action. That action might include honking their horn, giving the finger, blocking the other car in and shouting at the other driver, or even following them into the store to confront them. Their objective is no longer to find a parking spot, get that last ingredient, and get home to their guests. Instead, their objective is to ensure the other driver is punished and show the world they won't be a pushover.
Suppose a person has a higher level of differentiation. In that case, they have more time to think about the situation, recall their objective, and then decide what they want to do. The person may think, "Hey, this sucks and is annoying, but I still need to find a spot, get my item and get home." They don't have a visible angry reaction because they have no need for defensiveness. They don't need the other person to be bad. They may even have compassion and say, "Wow, that person must need to get to the store more than I do; maybe they have a sick child at home and needs to pick up some medicine."
Why is differentiation important in a family system? The more differentiated the parents, the less emotional upheaval and anxiety there is for their children. Suppose a child has a parent who gets angry frequently and blames other people, including the child. In that case, the child will believe they are responsible for their parent's upset. This can make them anxious, scared, reactive, and quick to assume blame as an adult. Children learn to manage their parent's emotions; if they can keep their parents stable and calm, they are doing a good job and are good. If they can't, then it's because they weren't good enough.