The greatest similarity, I think, is that the people who pray them believe the practice can be transforming. The most obvious difference, it seems to me, is that the first addresses Jesus and the latter do not. I do not know enough about the meaning and use of mantras in Buddhism to say much more.
While I don’t know much about the meaning and use of mantras in Buddhism, I have studied a bit about Buddhism, and I don’t think this is an entirely fair portrayal. In Buddhism, Nirvana is the “Absolute Truth” or “Ultimate Reality.” The liberated experience it, but human words cannot adequately describe it, so negative (apophatic) terms (“not this, not that”) are usually used instead.
Edward Conze has noted that when positive (cataphatic) language is used to describe Nirvana in Buddhist texts, they say that “Nirvana is permanent, stable, imperishable, immovable, ageless, deathless, unborn, and unbecome, that it is power, bliss and happiness, the secure refuge, the shelter, and the place of unassailable safety; that it is the real Truth and the supreme Reality; that it is the Good, the supreme goal and the one and only consummation of our life, the eternal, hidden and incomprehensible Peace.” (Quoted in Huston Smith & Philip Novak’s Buddhism: A Concise Introduction.) As one Theravadan scholar puts it: “Truth is not a result or an effect. It is not produced like a mystic, spiritual, mental state… TRUTH IS. NIRVANA IS.” Zen teacher Thich Nhat Hanh, similarly, calls Nirvana the “ground of being.” But apophatic terms are preferred, precisely because this reality cannot be comprehended.
The nearest analog for Nirvana in Christianity would be God. Apophatic language for God is also preferred by many traditional and orthodox Christian theologians, and especially in the Christian East.
In this light, Christians should not be excessively critical of Buddhism encouraging detachment from creatures in favor of Nirvana just because Nirvana is usually described apophatically. We encourage detachment from creatures in favor of God, and often describe God apophatically.