Given the existence of segregation in the south and the strength of it in the 1930's, it is arguable that changing the time of the story would affect the plot in a number of ways.
If the story were to be set in more modern times, particularly after the Equal Rights movement, one would imagine that the trial of Tom Robinson would be rather different. One would hope that the jury would be less biased against him simply because of the color of his skin.
It would also be significantly less impressive that Atticus would defend him so vigorously, it would be expected that a defense attorney would do their best to defend their client. It would also be far less likely that a group of men would gather to go down to the jail to try and lynch Tom Robinson as a matter of extra-judicial justice.
It might also be more difficult for Boo Radley to escape prosecution over the death of Bob Ewell. It is one thing for the Sheriff to have the type of discretion he did in Maycomb at the time but he likely wouldn't have the ability to simply let Boo off the hook, there would need to be a trial, etc.