The comparison with the Christian incarnation is unavoidable and is more misleading than helpful, and the differences are worth setting out plainly.
Number. Vishnu descends repeatedly, in every age and by some accounts innumerably. The Christian claim concerns a single event that is not repeated and could not be.
History. The Christian claim is deliberately anchored to a datable person under a named governor, and the tradition holds that if the event did not occur the faith is void. Avatars belong to cosmic time, to ages of vast length, and Hindu traditions have not generally made their salvific value depend on their historicity in the modern sense.
Constitution. An avatar is a form the deity assumes; Vishnu is not exhausted or altered by it, and remains fully himself in his own realm throughout. Christian orthodoxy insists on a permanent union of full divinity and full humanity in one person, and specifically rejected as docetism the view that the humanity was an appearance the divine put on. That rejected view is closer to some accounts of avatara than the accepted one.
Purpose. Avatars restore the moral order, defeat particular adversaries, and teach. Salvation in Christian terms is accomplished by a death and resurrection; nothing in the avatar scheme corresponds to atonement.
Within Vaishnavism the hierarchy is itself disputed. The Gaudiya tradition following Chaitanya reverses the usual order, holding Krishna to be the source of all divinity, svayam bhagavan, with Vishnu as his expansion rather than the other way round. Other schools rank the descents into full, partial and empowered forms. The Bhagavata Purana provides the material for both readings, which is one reason the argument has lasted.