Why do Orthodox Christians "cross themselves" different than Roman Catholics?

They touch their right shoulder first, then their left, whereas the Roman Catholics first touch their left shoulder. Is this difference important? Does it make any difference?

Orthodox cross themselves from right to left. first we will describe the mechanics of making the cross, then explain why it is indeed important that we make the sign of the cross correctly.

"Placing the cross on oneself"

1. We place our thumb and first two fingers together in a point, and our last we fingers flat against our palm. The three fingers together represent the Holy Trinity - Father, Son and Holy Spirit, and the two fingers in the palm represent the two natures of Christ.
2. We touch our forehead, then our belly, tracing the vertical part of the cross.
3. From our belly, we bring our hand up to our right shoulder, touching it.
4. We finish placing the cross on ourself by touching our left shoulder.

The act of "Placing the cross on oneself" is a request for a blessing from God. We make if from right to left to mirror the actions of the priest when he blesses us. The priest, looking at the parishioners, blesses from left to right. Therefore, the parishioners, putting on the sign of the cross on themselves, do it from right to left.

Because the Lord separated the sheep from the goats, putting the faithful sheep on His right side, and the goats on the left, the Church always treats the right side as the preferred side. We only cross ourselves with our RIGHT hand. The priest, when blessing a person, first touches or points to their RIGHT side, then their left. Also the censing of the Holy Table in the Altar is always done from the RIGHT side first; censing of the Ikonostasis, the Congregation and of the Church itself always begins with the right side. The priest always gives communion with his RIGHT hand, even if he is left handed. There are other examples of this right side preference.

When a parent makes the sign of the cross over a child, they will cross them from left to right, just as the priest blesses. When they make the sign of the cross over themselves, they would do it, logically, the other way.

The Catholic Encyclopedia states that in the Roman Catholic Church, the faithful crossed themselves from right to left, just as the Orthodox do, until the 15th or 16th century. They must explain why they have changed an ancient and apostolic tradition. We cannot answer as to their motivations.

Is it important to cross ourselves a particular way? In a word, YES. We do not have the authority to choose willy-nilly what parts of the Christina Tradition we want to follow. Our fathers, and countless saints crossed themselves from right to left. Ancient icons show Christ or bishops beginning a blessing from right to left. the right side is referred to in a preferential way many times in scripture and our sacred hymns What should we want to change?

Thanks for Fr Alexander Lebedev for providing the majority of the ideas in this answer.