Why Did the Analyst Revive Neo?

The Analyst (Neil Patrick Harris) reveals that he revived Neo and Trinity to power a new version of the Matrix built on desire and fear — a desire for the things people don't have and fear of losing the things that they do.

Why did the analyst keep Neo alive?

The Analyst is the new Architect.

When this happened, a program called The Analyst observed that Neo was an anomaly like none other and that Neo's choices were based on his love for Trinity, a companion that the earlier anomalies did not have. So he resuscitates both of them.

Why did Agent Smith save Neo?

By saving Neo and Trinity, Smith ensured his own continuing survival, and ability to exist as he sees fit within the Matrix. Since Smith proclaims that he and Neo's unlikely alliance is now over though, one assumes Neo and Trinity haven't seen the last of their old nemesis.