People often conflate empathy as caring. Empathy does not mean someone cares. Here are the definitions of each word
Empathy is the act of recognizing the emotional state of another and then internalizing that emotional state. You are then mirroring the other person’s emotional state. It doesn’t mean you care, it doesn’t mean you don’t care. Empathy is just about mirroring from a terminology standpoint.
Sympathy. You are rationally looking at the situation to understand what is going on. You can have feelings about the situation. You can empathize with the other person. You can even care about what’s going on. But at the basic of what Sympathy means, you are taking a step back from the emotions to rationally look at the situation.
Caring. Caring is yet another separate step. You are actively trying to mitigate the situation. You may chose to mitigate by showing empathy. Giving the other person comfort in knowing you can feel the same way about the situation. You can chose sympathy, rationalizing the situation and saying, there’s something bigger going on that we all need to consider. You can be motivated in many ways, duty, greed (it will help you look good to others), because of moral standing. etc. Caring is the act of making the choice to do something positive regardless of motivation.
But just because someone is empathizing with you, doesn’t mean they care. Narcissists use this against you. They empathize to make you think they care about you. When in reality they are manipulating you.
Relying solely on empathy, you end up pushing your emotional interpretation onto others. This is often referred to as being a Karen. It’s based on the belief that “I feel this way about this situation, so others must feel the same way” without sympathy, you get trapped by arrogance in believing you are seeing the world as others see it. You have to take a step back to make sure you are correct.
Sympathy has it’s own problems. People see you as cold or callous because you are not expressing emotional empathy. Even when you do but contradict what that emotion says to do. People still think of you as being mean.