So I'm a bit confused. What's the difference between penance and penitence? Are they essentially the same thing? Thank you everyone.

From Dictionary.com

pen ance   
–noun
1. a punishment undergone in token of penitence for sin.

pen i tence   
–noun
the state of being penitent; regret for one's wrongdoing or sinning; contrition; repentance.

As you can see, a penance is a punishment; penitence is a feeling of guilt or being sorry.