Sympathy may have its place in compassion, but sympathy alone is not compassion. The word sympathy comes to us from the Greek language. To sympathize with someone is to share in that person's feeling or feelings. Certainly if we are closed off to the feelings of others, it will be hard to be of any help to them. Compassion is more than sympathy.
Compassion is a Latin derivative.
It means to share more than just the feelings of another person. It comes closer to meaning that one suffers with another. This puts feeling much deeper in the body. The ancients felt compassion in their gut. It is not just some head trip.