How are prayer and asking the saints to intercede distinguished?

A Catholic answer to your question would be to say that "pray" has more than one sense.

First, it means simply "ask", as you suggest. This is very obvious if we look at older styles of English. For example, from Hamlet:

Speak the speech, I pray you, as I pronounced it to you (Act 3, Scene 2)

Here, "pray" means "ask". "I pray you" ends up basically meaning "please". This is the sense used when referring to the intercession of the saints.

Then it has a deeper meaning. Some quotes from the Catechism of the Catholic Church:

Prayer is the raising of one's mind and heart to God ... Whether we realize it or not, prayer is the encounter of God's thirst with ours. ... Prayer is the response of faith to the free promise of salvation and also a response of love to the thirst of the only Son of God. ... Christian prayer is a covenant relationship between God and man in Christ. It is the action of God and of man, springing forth from both the Holy Spirit and ourselves, wholly directed to the Father, in union with the human will of the Son of God made man. ^{(CCC, ¶¶2559ff.)}

This is a deeper meaning, referring to the covenant, communion relationship between human beings and God. Intercession, worship, penitence, and joy (and much else!) are part of the life of prayer. It is the expression and mode of the relationship between the human being and God, and in that way it is exclusive to God.

But the key point about the Catholic idea of prayer is that it isn't primarily individual as in more Protestant understandings. Rather, it is an ecclesial task, a task in which the whole Church participates together. This is seen supremely in the offering of the Mass:

To the offering of Christ are united not only the members still here on earth, but also those already in the glory of heaven. In communion with and commemorating the Blessed Virgin Mary and all the saints, the Church offers the Eucharistic sacrifice. In the Eucharist the Church is as it were at the foot of the cross with Mary, united with the offering and intercession of Christ. ^{(CCC, ¶1370)}

The saints are part of the Church. (The traditional "parts" of the Church are the Church Militant here on earth, the Church Suffering in purgatory, and the Church Triumphant in Heaven.^{(CCC, ¶962)}) So we join in the prayer of the whole Church to God. In this context, we are brothers and sisters with those making their prayer in Heaven. And what could be more natural than asking their prayer for us?

Prayer is a lot more than asking, yet "pray" also means "ask". It has a different sense when referring to God to what it means in referring to God.

As to the verse in Deuteronomy, to my mind this is a completely different matter to intercession to the saints. Let's look at the passage in its context:

No one shall be found among you who makes a son or daughter pass through fire, or who practises divination, or is a soothsayer, or an augur, or a sorcerer, or one who casts spells, or who consults ghosts or spirits, or who seeks oracles from the dead. (Deuteronomy 18.10f., NRSV)

The context of the prohibition clearly refers to sorcery and magic, asking the dead about the future. This is the behaviour of Saul in consulting the Witch of Endor, for example. Catholics do not say, "Mary, who should I marry?" and expect a lightning bolt answer. Instead, they would say, "Mary, by your intercession help me to determine who I should marry." They are very different things.

Moreover, there is a significant difference between the time of Deuteronomy and the modern day. The idea of an afterlife was very different in ancient Judaism to the beliefs held today. (Remember that the distinction between Pharisees and Sadducees in Jesus' time was the former's belief in a resurrection.) According to the Catholic tradition, until the Resurrection of Christ the dead were trapped in Hell, and by his Resurrection he has led them to Heaven (the Harrowing of Hell). It is precisely the saints in Heaven of whom Catholics ask prayer. Before the Resurrection there were none, so prayer to the dead was prayer to souls in Hell.^{(See CCC, ¶¶631ff.)}