In the Catholic church, the people can have Mass celebrated for someone. From the salvation point of view, what is the difference between celebrating a Mass:
Based on this difference, is it possible to say which purpose is more important?
When Catholics say that a Mass is celebrated "for" something or someone, it just means that the priest offers the Mass—which for Catholics is the very highest form of prayer—for the sake of some intention. It is a kind of intercessory prayer, and Masses can be offered for any kind of intention (for example, for an end to a war or famine, for a recovery from illness, or simply for the sake of the local Church).
When they say that the Mass is offered "for someone," it just means that the priest offering the Mass at the behest of that person (or perhaps even a third party; for instance, a mother might ask a priest to offer the Mass for her son).
When that person in question is deceased, the Mass in question is usually offered for the repose of the soul of that person.
(For those who are not familiar with the doctrine, the Church teaches that it is possible for someone to die in the state of friendship with God, but not completely sanctified or purified. For those persons, a purification is needed after death: what Catholics call "Purgatory." Note that Purgatory is not an intermediate state between Heaven and Hell, but, as it were, the antechamber of Heaven, where people prepare to enter Heaven. According to Catholic doctrine, all of the souls in Purgatory will reach the Divine Presence in the end.)
Catholics believe that intercessory prayer can help those persons that need it to complete their purification after death. Since the Mass is the highest form of intercessory prayer possible, many (if not most) Masses are celebrated with for the sake of some loved one who is deceased.
Returning to the question, determining which purpose is the more important is complex, because it depends on the intention requested.
For example, the repentance of someone not in friendship with God is more important than the purification of a deceased person in Purgatory, since the salvation of the souls in Purgatory is assured, whereas the salvation of the former person is in peril.
On the other hand, praying for someone in Purgatory is probably (at least absolutely speaking) more important than praying for someone's physical healing, because (at least according to the testimony that many saints and mystics have given) the purification in Purgatory causes a suffering much greater than that caused by illness.
One should also keep in mind that although each Mass has what is called a "first intention"—which is regulated by Canon law to avoid abuses in collecting donations for Masses—a single Mass can have any number of "second intentions." Therefore a single Mass could be offered for both a living person and a deceased one, or several of each. Moreover, the "second intentions" are not worth "less" than the first; it is just that only first can be attached to a donation (and priests may collect a maximum of one per day).
The reason that most Masses are offered for deceased souls (as a first intention) has less to do, I think, with the intrinsic importance of the intention, as with the spiritual needs of the faithful. Many people request Masses for their deceased relatives, usually with a small donation (stipend), and therefore these, as a practical matter, become the most common first intention.
You're essentially asking "What's the difference between praying for the dead and praying for the living; and which is more important?"
Catholicism, by its interpretation of the communion of saints, believes that each living believer is in communion not only with all others living, but equally with those who have "fallen asleep in Christ":
"So it is that the union of the wayfarers with the brethren who sleep in the peace of Christ is in no way interrupted, but on the contrary, according to the constant faith of the Church, this union is reinforced by an exchange of spiritual goods."
(Catechism of the Catholic Church, paragraph 955, quoting Lumen Gentium, section 49)
Lumen Gentium continues:
Fully conscious of this communion of the whole Mystical Body of Jesus Christ, the pilgrim Church from the very first ages of the Christian religion has cultivated with great piety the memory of the dead, and "because it is a holy and wholesome thought to pray for the dead that they may be loosed from their sins" [2 Maccabees 12:46], also offers suffrages for them.
(Lumen Gentium, section 50)
In this sense, there is no difference between praying for the living and praying for the dead—except, as you point out, that to pray for the living is usually to pray for their temporal well-being, whereas to pray for the dead is to pray for their spiritual well-being.
Now, as far as "which is better?" I don't know that there is an answer. Aquinas does talk about whether we ought to pray for others, and concludes,
When we pray we ought to ask for what we ought to desire. Now we ought to desire good things not only for ourselves, but also for others: for this is essential to the love which we owe to our neighbor.
(Summa Theologica, Second Part of the Second Part, Question 83, article 7)
He does not similarly discuss whether we ought to pray for the dead, though he does have a whole Question (Summa Theologica, Supplement to the Third Part, Question 71) dedicated to other questions about the prayers for the dead, including Masses offered for the dead. He nowhere says, nor can I find anything in the Catechism or in a quick search through papal encyclicals and other doctrinal communications, that the one sort of prayer is more important than the other. My opinion is that neither is more important; however, I have no sort of authoritative backing for that.
A Different Approach
Under VIRTUES AND VICES in A CATECHISM OF CHRISTIAN DOCTRINE | CATHOLIC TRUTH SOCIETY, Q.3221 reads:
322. Which are the Seven Spiritual Works of Mercy?
[...]
7. To pray for the living and the dead.
Therefore, both ought to be prayed for, and offering Mass is the best way to pray for the living and the dead.
MattGutting's and AthanasiusOfAlex's answers have done a good job explaining some aspects of the Catholic practice and basis.
Catholics are familiar with the following Fatima Prayer:
'O my Jesus, pardon us, and save us from the fire of hell; draw all souls to heaven, especially those most in need.'
Thus to answer - [b]ased on this difference, is it possible to say which purpose is more important? - I would say the most important would be praying for the one most in need. Thus to me, praying more and with greater intensity (mass, rosary, memorares, etc.) and offering sacrifices and mortifications for a family member, a friend, or a colleague languishing in a gravely sinful life would rank above the holy souls in purgatory whose salvation is assured.
Similarly, with the living, my supplications would focus more on the sick or gravely injured, the unemployed or those in some other material need among those whom I love and I ought to pray for, whilst still saying the customary prayers for others not in their condition.
On the other hand, it is easy to forget to pray for the holy souls in purgatory, among whom may be a family member (natural or Christian e.g. priests and religious in one's life, godparents, etc.), a friend, or a former colleague. Since the holy souls cannot help themselves, in this sense the holy souls in purgatory are most in need.
Perhaps rather than focusing on which is more important, it behooves us to continually pray for the living and the dead without expecting a reward, purely out of love for the LORD and for others for his sake. The LORD is generous, we will be blessed, because [some of those for whom we pray for] cannot repay us. We will be repaid at the resurrection of the just2.
"[S]o whoever strives for his neighbor's salvation first of all profits himself and afterwards his neighbor." - St. John Damascene quoted in Supplement to the Third Part, Question 71, Article 4, contra | Summa Theologica.
1. cf. Q.322, Virtues, Gifts of the Holy Spirit and the Counsels | PENNY CATECHISM.
2. cf. Lk 14:13-14.
Endnote
Please note that when praying for the living and the dead, I wrote, family, friends, and [former] colleagues, therefore pointing out that our prayer for others ought to be start with those God has placed by our side beginning with those connected to us by ties of blood and christian faith. For example:
Prayer for Others
O Jesus, have mercy on your holy Church; take care of it.
O Jesus, have pity on poor sinners, and save them from hell.
O Jesus, bless [my spouse and my children], my father, my mother, my brothers and sisters, and all I ought to pray for, as your Heart knows how to bless them.
[O Jesus, bless the Pope, the bishops and the priests.]
O Jesus, have pity on the poor souls in purgatory and give them eternal rest.
- Source - A Simple Prayer Book | Catholic Truth Society
