In Catholicism, do scripted and unscripted prayers have equal importance to one's spiritual identity and development?

I don't quite know how you would measure the "importance" of a prayer in the first place. There are really several ways of looking at it:

Are we talking about importance to the believer?

Importance to God?

Importance to the congregation?

Importance as a historical event?

In terms of importance to the believer, I think this is the one case where we can say that it's hard to imagine a believer who doesn't have any form of unscripted prayer life whatsoever. So in this case, we can argue that unscripted prayer is more important to the believer, purely on the grounds that it is an indicator that they are a believer in the first place (even this is a bit shaky though - I'm not trying to condemn anyone whose prayer life is 100% scripted to the status of "non believer")

For all the other ways of talking about importance above though, I don't think the type of prayer (unscripted/scripted) makes any significant contribution to it's importance in itself.

What I do think though is that these two types of prayers have different functions. Public scripted prayers where the congregation participate have a social element, and there's a sense in which they contribute to the social cohesion of the church, and doctrinal unity, rather than purely being expressive of a particular thought or desire towards God.

To a degree this can also be the case for unscripted public prayers (whether in a whole congregation- depending on your church, or in a small prayer group), but in the case of a private prayer between an individual and God, there isn't the same social aspect between fellow believers.

Neither one is necessarily better than the other, mind. The social aspect can be detrimental, such as in the case of an unscripted public prayer where the speaker is actively using the prayer to show off their theological knowledge/vocabulary or character, or in the case of a scripted prayer which people go along with but aren't really sincere in it.

In part this answer is based on what I've learned about the philosophy of society by listening to John Searle lectures - he makes the claim that a lot of communal religious practices serve two functions (in the christian viewpoint we would view both of these as purely natural functions, and also add a supernatural function to Christian practices - Searle doesn't do this as he is an atheist). One function is the mere purpose of the action in itself - for example, a tribe with a practice of rain dancing, performs the rain dance in order to try and cause rain to happen, but there is also a second function in that the ritual improves social cohesion by the communal practice of beliefs.

Christian practices such as liturgy also have this function of social cohesion. We don't want to reduce them to the naturalistic level where that is the only function or benefit, but it is a real benefit nonetheless, especially for the person's spiritual identity. Part of any healthy Christian's spiritual identity is being part of the church.

This answer probably sounds rather bland and unspiritual - I can't really comment much from a Catholic perspective in terms of how they view scripted prayers, only how I generally think they benefit believers regardless of denomination.

Short answer: yes.

The point is to pray, or as the Apostle Paul exhorts the faithful, to pray continually. (1 Thessalonians 5:16-18).

16 Rejoice always, 17 pray continually, 18 give thanks in all circumstances; for this is God’s will for you in Christ Jesus.

Prayer as God's gift

CCC 2559 "Prayer is the raising of one's mind and heart to God or the requesting of good things from God." But when we pray, do we speak from the height of our pride and will, or "out of the depths" of a humble and contrite heart? He who humbles himself will be exalted; humility is the foundation of prayer, Only when we humbly acknowledge that "we do not know how to pray as we ought," are we ready to receive freely the gift of prayer.

It isn't so much "what" you pray but "how" you pray. Pray from the heart.

CCC 2562 Whether prayer is expressed in words or gestures, it is the whole man who prays. But in naming the source of prayer, Scripture speaks sometimes of the soul or the spirit, but most often of the heart (more than a thousand times). According to Scripture, it is the heart that prays. If our heart is far from God, the words of prayer are in vain.

This encouragement to pray in all circumstances gets emphasis in the Catechism.

CCC 2659 We learn to pray at certain moments by hearing the Word of the Lord and sharing in his Paschal mystery, but his Spirit is offered us at all times, in the events of each day, to make prayer spring up from us. Jesus' teaching about praying to our Father is in the same vein as his teaching about providence: time is in the Father's hands; it is in the present that we encounter him, not yesterday nor tomorrow, but today: "O that today you would hearken to his voice! Harden not your hearts."
CCC 2660 Prayer in the events of each day and each moment is one of the secrets of the kingdom revealed to "little children," to the servants of Christ, to the poor of the Beatitudes. It is right and good to pray so that the coming of the kingdom of justice and peace may influence the march of history, but it is just as important to bring the help of prayer into humble, everyday situations; all forms of prayer can be the leaven to which the Lord compares the kingdom.

The Church teaches that prayer, when humbly offered from an open heart, is a form of (spiritual) communion.

Prayer as communion
CCC 2565 In the New Covenant, prayer is the living relationship of the children of God with their Father who is good beyond measure, with his Son Jesus Christ and with the Holy Spirit.
The grace of the Kingdom is "the union of the entire holy and royal Trinity . . . with the whole human spirit." Thus, the life of prayer is the habit of being in the presence of the thrice-holy God and in communion with him. This communion of life is always possible because, through Baptism, we have already been united with Christ.

To sum up: prayer isn't about importance, it is about reaching toward God from our hearts. Worrying about "importance" misses the point of prayer.

Longer answer: the Lord's Prayer provides a perfect model of prayer for the faithful

The Catechism covers prayer in over 300 articles. (CCC 2558 - 2865). I'd recommend reading it from first to last to get an overview of the teaching of prayer. A wealth of references to prayer from scripture, saints and theologians are included in foot notes that flesh out the understanding of prayer as a key element of faith. The Lord's Prayer has a detailed section to finish off the Catechism. (Ending on a high note, as it were).

CCC 2766 But Jesus does not give us a formula to repeat mechanically. As in every vocal prayer, it is through the Word of God that the Holy Spirit teaches the children of God to pray to their Father. Jesus not only gives us the words of our filial prayer; at the same time he gives us the Spirit by whom these words become in us "spirit and life." Even more, the proof and possibility of our filial prayer is that the Father "sent the Spirit of his Son into our hearts, crying, 'Abba! Father!'" Since our prayer sets forth our desires before God, it is again the Father, "he who searches the hearts of men," who "knows what is the mind of the Spirit, because the Spirit intercedes for the saints according to the will of God."

As above, it isn't just the words of the prayer, it is in opening one's heart up in prayer that prayer becomes the great gift from God.

CCC 2611 The prayer of faith consists not only in saying "Lord, Lord," but in disposing the heart to do the will of the Father. Jesus calls his disciples to bring into their prayer this concern for cooperating with the divine plan.

Some might conclude from the description "perfect" that The Lord's Prayer is more important. It is certainly the great gift of prayer given by Jesus himself to his people.

The Lord's Prayer is the most perfect of prayers.... In it we ask, not only for all the things we can rightly desire, but also in the sequence that they should be desired. This prayer not only teaches us to ask for things, but also in what order we should desire them. ~ St. Thomas Aquinas, Summa Theologica, II-II, 83, 9.

But the Church does not teach that other prayers are less important. It teaches that prayer is important, even crucial, to Christian life.

"THE SUMMARY OF THE WHOLE GOSPEL"

2761 The Lord's Prayer "is truly the summary of the whole gospel." "Since the Lord . . . after handing over the practice of prayer, said elsewhere, 'Ask and you will receive,' and since everyone has petitions which are peculiar to his circumstances, the regular and appropriate prayer [the Lord's Prayer] is said first, as the foundation of further desires."

You could call the Lord's Prayer the gateway prayer that opens up infinite doors in one's heart to God. (On an experiential note, when I pray I most often start with the Lord's Prayer to get myself in a good prayerful state. It is a great start to any longer period of praying. Other times, particularly when attending Adoration, my opening prayer is an extemporaneous salutation to Jesus as I sit in front of the Body of Christ, exposed on the monstrance).

CCC 2762 After showing how the psalms are the principal food of Christian prayer and flow together in the petitions of the Our Father, St. Augustine concludes:

Run through all the words of the holy prayers [in Scripture], and I do not think that you will find anything in them that is not contained and included in the Lord's Prayer.

The church teaches that prayer is important in spiritual development, and spiritual identity as one who is with God through Jesus Christ and the Holy Spirit. The point is to pray. Quibbling over "importance" rather misses the point, as the disciples seemed to be missing the point in Luke 22:24-30.

24 Now there was also a dispute among them, as to which of them should be considered the greatest. 25 And He said to them, “The kings of the Gentiles exercise lordship over them, and those who exercise authority over them are called ‘benefactors.’ 26 But not so among you; on the contrary, he who is greatest among you, let him be as the younger, and he who governs as he who serves. 27 For who is greater, he who sits at the table, or he who serves? Is it not he who sits at the table? Yet I am among you as the One who serves.

28 “But you are those who have continued with Me in My trials. 29 And I bestow upon you a kingdom, just as My Father bestowed one upon Me, 30 that you may eat and drink at My table in My kingdom, and sit on thrones judging the twelve tribes of Israel.”

Come to the table, it doesn't matter where you sit.

Pray with your heart.