Thanks for asking!
First of all, penance is sort of confusing, for has two meanings. It can refer to the sacrament of penance, also known as confession and reconciliation. But it is usual easy to tell if that meaning is being used.
Sort of connected to that definition is the penance that a priest assigns a person who is receiving the sacrament of penance. In the early days of the Church, when there was public confession, there was public penance also, such as having to stand by the Church door every day for a year(or much less; I'm not to up on the subject actually), asking those who go in to pray for them. As time went one, the Church came to emphasize the mercy of God to a greater degree than the justice of God when it celebrated the sacrament of penance. It was made easier for a sinner to repent, expressing God's desire that all repent and come to Him.
Penance is still a part of the sacrament of penance; now it takes the form of prayers the priest assigns to the person, instead of more public acts.
About the other type of penance, the type you take by your own free will, it is attached to the Catholic view of suffering i.e., that you can offer it up to God, for his glory, and for the benefit of yourselves and others. Now, if that be the case, then suffering that you choose to take on would be no different, so long as your motives are pure, not wanting the respect of men, and so forth, as Jesus says in the sermon on the mount, when he discuses fasting.
I'm sorry I was so brief, I have to leave shortly; I'll try to post more later.
