Why is Tom Robinson's sentence important?
Tom Robinson's sentence is never actually revealed in the book. A sentence is the term in prison or the announcement of the punishment. We do know he was taken to a prison because we hear second hand about how he was shot at.
I assume you are asking about the verdict (the decision the jury came to) which was guilty. The other editors have addressed that. If you do indeed mean the term sentence, I think what's important about it is that he never got to serve it. Before there was even a chance for appeal, before the sentence was revealed, Tom was killed. I guess you could say that his sentence ended up being the death penalty unjustly applied. This is important because it demonstrates the deep seeded racism at work in the South in the 30s.
In regards to the verdict, there is one thing that is significant for this time period. The all white jury does find him to be guilty, which is really no surprise. The importance of their verdict is that the jury actually considered the case before immediately finding him guilty. The jury deliberated for an unusually long period of time, well into the late evening. The fact that the certain jurors needed convincing before immediately subscribing to a guilty vote shows progress in this largely racist community. Some interpret this as progress toward the equal treatment of African-Americans in a criminal case for the Maycomb community.
A not guilty verdict in the Tom Robinson trial would have been an extraordinarily rare event in 1930s Alabama. As Atticus pointed out in To Kill a Mockingbird, no one could really expect an all-white jury to take the word of a black man (Tom) over that of a white man (Bob Ewell). Atticus realized he had little chance of acquittal at trial, but he hoped to eventually have the verdict overturned on appeal. Had Tom been found innocent of the charges against him, it would have been a major step in racial justice for a Deep South state.
In the book To Kill a Mockingbird Tom Robinson's sentence is important because he is an innocent man who knows that he ahs been condemned to prison or death. As he awaits sentencing he decides that it is better to take his own life than o live a life of misery in an institution or on a chain gang.
Tom does not trust the people who will sentence him anymore than he trusts them for finding him guilty. They can not see through their racism to make a fair decision.