To excuse is to release from blame, or from a duty or obligation. It need not have any connection to a transgression, a sin, or wrong action of any sort. For instance, when I was a child, I would ask my parents after I had finished dinner, "May I please be excused". It just meant I wanted permission to leave the table. To pardon is to release from blame. Both of your examples involve transgressions, so either word works. However, in an example like mine at the dinner table, only "excuse" works.

I would say "excuse" is more commonly used in everyday speech. In both examples I would use "excuse". Pardon to me is a bit more formal/old-fashioned for example people would be pardoned by the king/queen in the past. Or in a more official capacity for example presidents can "pardon" someone that went to prison.