I agree with Matt Gutting, and I would add that the direction of focus is an important difference. Disclaimer: I am answering from the Catholic perspective here, and my knowledge of other religions is not extensive.
Edit, in response to people asking for a source.
Catechism of the Catholic Church, paragraph 2708 reads: Meditation engages thought, imagination, emotion, and desire. This mobilization of faculties is necessary in order to deepen our convictions of faith, prompt the conversion of our heart, and strengthen our will to follow Christ. Christian prayer tries above all to meditate on the mysteries of Christ, as in lectio divina [meditative reading of Sacred Scripture] or the rosary[emphasis added]. This form of prayerful reflection is of great value, but Christian prayer should go further: to the knowledge of the love of the Lord Jesus, to union with him[emphasis added]. (516, 2678)
The Rosary is meant to focus on God. When a Catholic is "meditating on the Mysteries of the Rosary," he or she is thinking deeply about and objective truth that exists outside of his or her own mind. The goal of this exercise is not the change of one's own mental state, but rather furthered knowledge of and love for God. ("Spiritual Awareness" is not synonymous with this goal, because Catholics understand spirits to be either the Holy Spirit, or angelic, or demonic. Thus, a general ability to perceive and receive influence from spirits or a sense of "being spiritual" is regarded as a liability or a vulnerability, because demonic spirits could potential appear disguised as God or angels.) Edit, thanks to Matt for a great comment: "Spiritual Awareness" could mean the condition, not of being aware of many spirits, but of being aware of the condition of one's own soul. But awareness of the condition of one's own soul is also not a goal of Christian prayer. 1 Cor 4:3 says "...I do not even pass judgment on myself" and Christ's constant exhortations in the Gospels to "remain vigilant" lead to the teaching that we do not necessarily know, nor do we seek to know, the exact condition of our own soul. We should examine our conscience, and repent promptly of any sin, but self-awareness gained of long introspection is not a goal of the Christian life.
For a Catholic, peace of mind is incidental, not essential, to the practice of meditating on the Mysteries. For example, a lack of inner peace, a sense of distraction, or a sense of dryness or distaste are not regarded as flaws in the meditative technique, but are rather regarded as an invitation by God to continue praying, and to gain more merit from it for having persevered through a lack of interior consolation. So there is a secondary difference: That in the Catholic Rosary, interior mental trials are to be endured (though willful distraction should be avoided), while I believe that mantras are meant to combat directly such interior disquiet. A prolonged period of spiritual desolation in prayer is regarded as an ordinary part of the spiritual life, and is not considered a consequence of the proficiency in meditation of the person who experiences it. (See this article from the online Catholic Encyclopedia on the states or ways of the spiritual life:
"In the works of St. John of the Cross these purifications are called nights, and he divides them into two classes, the night of the senses and the night of the spirits. In the state of beginners the soul is often favored by God with what are called "sensible consolations" because they have their beginning and are felt chiefly in the senses or sensible faculties. They consist in sensible devotion and a feeling of fervour arising from the consideration of God's goodness vividly represented to the mind and heart; or, from external aids, such as the ceremonies of the Church. *These consolations are often withdrawn, and a state of desolation ensues, and then the passive purification of the senses begins.* [emphasis added]"