The dictionary descriptions of compassion and pity are similar. But in modern English usage, pity has an element of holding yourself superior to whatever conditions are affecting the person you feel pity towards.
So you may feel sorrow for someone and pity them, but at the same time feel they are in some way responsible for their predicament. It's a subtle thing.
Compassion and sympathy are the words used to express the sorrow or concern you feel for someone when you don't have this same feeling of superiority or feeling that the person is somehow to blame for their misfortunes.
An example of this is that to say to someone, "I pity you" is quite an insult while to say to someone "I sympathize with you" or "You have my compassion" is comforting or reassuring.
I've seen karuna described as compassionate pity. And with the qualifier of "compassionate" it works around this subtle element of superiority/blame.
None of these words necessarily mean taking action to help or relieve the suffering of the being who has prompted your compassion, pity, or sympathy. You may feel moved to help or you may feel unable to help. Your decision to help, or not, is different than the feeling of sorrow for another being. An example: two people watch a documentary on drug addiction. One feels pity for the addicts as they believe them to be responsible for their predicament. The other feels compassion for the addicts as they view addiction more as an illness. Neither one takes any action to help but they have each experienced the emotion of sorrow for another being in different perspectives.
From a Buddhist point of view, feeling superior to someone else (by looking on their misfortunes with pity instead of compassion) would be short sighted as the ripening of our own karmic fruit is always unknown until it happens.
It's possible this view of pity, as being sorrow touched with superiority to that which is being pitied, is not universal. However here in the United States, if a person were to say, "you have my pity" an expected response might be, "I don't need your pity!" or "I don't want your pity!". Actually those words are said even in cases where a person suspects they are being pitied. In these parts, "pity" definitely has negative aspects which are not present in "compassion" or "sympathy".