Use this expression when you want to talk about two different things. The first thing is big or important. However, the second thing is even bigger or more important. Here are some examples:

She's not just super smart, but also hard-working and easy to get along with.

Kids who learn martial arts gain not just strength and fighting skill, but confidence as well.

Another phrase that's used in the same way is "not only ___, but ___":

You not only disobeyed me, you betrayed my trust.