Atticus had been planning to appeal the verdict even if this had little chance of success. However, in Chapter 24, we learn that Tom is shot and killed trying to escape. It is clear that Atticus and Tom both knew the slim chance the appeal would work. Upon learning of how Tom was shot while trying to escape, Atticus says, "I guess Tom was tired of white men’s chances and preferred to take his own." Tom knew the appeal was a long shot. He knew his previous trial was a long shot because of the racist thinking of the jury. He decided to take his own chances. 

In Chapter 25, Scout adds that the people (white) of Maycomb were interested in Tom's death for only two days. To them, he was just another black man, a statistic of the lowest class of people. Most of them get on with their lives, thinking nothing of the injustice. 

At the end of Chapter 25, Scout reads Mr. Underwood's editorial which compared Tom's death to "the senseless slaughter of songbirds." Mr. Underwood is one of the few in Maycomb who speak to and stand for justice. Scout concludes that Tom Robinson never had a chance. 

Then Mr. Underwood’s meaning became clear: Atticus had used every tool available to free men to save Tom Robinson, but in the secret courts of men’s hearts Atticus had no case. Tom was a dead man the minute Mayella Ewell opened her mouth and screamed. 

The men of the jury outwardly swear that they will make a just decision, but in the "secret courts" of their hearts there is a racism they refuse to overcome. Hence, they convict Tom even though the evidence shows his innocence.