Compassion is feeling for someone else’s distress and wanting to alleviate it.

Mercy is an act of leniency.

I think compassion is more important and that you cannot have mercy without it.

That mercy doesn’t require compassion but compassion requires mercy.

Compassion is a feeling of understanding and mercy is a judgment rather than a feeling, although it is based on the feeling of compassion. Is that the opposite of what @Blondesjon said? (now that I read his post again..)

I keep getting excited when I see the Mouse avatar and then saying, “Oh darn!”

Compassion is a personal response.
Mercy is a social decision.

This is just an impression but it feels like compassion comes from a position of equality while mercy comes from a superior person or institution.

To me,and this may be just my perception:
Mercy is something given even when the offending party hasn’t earned a break.
Compassion is when you feel empathy and give willingly.

So, mercy is something given that isn’t earned (a pardon granted to someone unworthy), compassion is seeing someone’s need/weakness/pain and helping them out because you want to or are compelled to help.

mer·cy

/ˈmərsē/

Noun

1.Compassion or forgiveness shown toward someone whom it is within one’s power to punish or harm: “the boy was begging for mercy”.
2.An event to be grateful for, esp. because its occurrence prevents something unpleasant or provides relief from suffering.

Synonyms

pity – compassion – clemency – grace – charity – ruth

Mercy; don’t chop their head off, whether they deserve it or not.

Compassion; chop their head off, but feel sorry about it, and not hate them. Whether they deserve it or not.

It seems to me that mercy implies authoritarian relationships, as it’s the person (or even our imagined gods) in a position of power who can be merciful to someone subordinate.

Mere compassion has a more communal and egalitarian connotation and is more synonymous with the feeling of empathy.