From my experience, metta is when you wish good things for others and karuna is when you understand people’s present or future suffering. If someone is suffering, you can feel compassion. If you know someone is going to suffer in the future, you can also feel compassion for them. You don’t have to necessarily feel sad for others, but you have to understand their sufferings and sympathize with them. For example, just as I don’t want to suffer, others don’t want to suffer as well.
I find that when I practice compassion (karuna) meditation, I become happy.
The challenge is to know when you are practicing loving kindness and when you’re practicing compassion. I used to mix the two in the past. When I practiced compassion meditation, it usually turned into loving kindness meditation in the past.
The phrase “May this person be free from suffering” can be meditated upon with either loving kindness or compassion. If you focus more on wishing for a person’s welfare and happiness, it’s loving kindness. If you focus more on understanding their suffering and sympathizing with them, it’s compassion.
This is how I understand it. I hope this is helpful. If anyone would like to say anything related to this based on the scriptures or their personal experiences, please feel free to do so.