What are the reasons that the prince decided to sentence Romeo to exile instead of death?
As the previous post points out, there is no overt statement offered Shakespeare's part as to why the Prince does not have Romeo executed. Nevertheless, there are implications in the narrative of the play, such as the fact that Mercutio, who is Romeo's friend, is a relative of the Prince.
Another possible explanation for the Prince's not exacting his earlier pronouncement of death to those whom he has told
If ever you disturb our streets again,
Your lives shall pay the forfeit of the peace (1.1.70
is that the Prince feels that demanding Romeo's death may cause even more feuding and killing. For, Romeo has a reputation of being a peaceful man, according to Lord Capulet, so his act demands cause.
These are two reasons for which the reader can find substantiation in the text, an essential to proving one's response. Certainly, opinion never has any argumentative powers or verity in responding to objective questions in literature.
We are not really actually told why the prince decides to exile Romeo instead of having him killed. The prince himself had said that any more fighting would be punished by death. So we have to guess. I assume that the prince exiles Romeo because he was obviously provoked into killing Tybalt and (possibly) because Mercutio was a relative of the prince.
If you think about it, it would be pretty harsh to kill Romeo for what he did. By the law, Tybalt was supposed to die for killing Mercutio. So why should you kill Romeo for killing Tybalt? But I imagine that the prince couldn't just let Romeo go completely free because the Capulets would get really mad. So I think he compromised and just exiled Romeo.