For Catholics, the Mass is the Lord's Supper and the Lord's Supper is the Mass. They are interchangeable.
The difference is in how Protestants and Catholics understand it.
For Catholics, Mass is the sacrifice of Christ re-presented under the signs of bread and wine. It is the same sacrifice that took place on the cross; the difference is in how the sacrifice is presented. On the cross, the sacrifice was earthly, fleshy and bloody. In the Mass, the sacrifice is not-earthly, it is glorified and it is not bloody and fleshy. The manner in which the sacrifice was presented on the cross lead to the suffering and death of Christ. This is because its was presented in its earthly form. In the Mass, the sacrifice does not lead to the suffering and death of Christ because it is not earthy, but is glorified. It is also not bloody and fleshy; it is presented in Sacramental form under the signs of bread and wine. Christ cannot die again. The resurrection is what makes the Mass possible. Without the resurrection, there could be no Mass.
Does the Mass "expiate man's sins and the punishment for those sins?" In a general sense, yes. The other Sacraments flow from the Mass. The other Sacraments, therefore are extensions of the Mass. They are the particular way the graces Christ won are applied.
For Protestants, whatever the Lord's Supper is, whatever they believe about it, it isn't what I said above.