The jury passed a verdict of guilty, with an appeal to the judge for clemancy. と The jury passed a verdict of guilty, depite an appeal to the judge for clemancy. はどう違いますか？説明が難しい場合は、例文を教えて下さい。

There’s a slight difference in meaning and intention between these two.
“The jury passed a verdict of guilty with an appeal to the judge for clemency” means the jury found the defendant guilty AND also asked the judge to show clemency, meaning leniency or mercy, likely in the punishment/sentencing.
“The jury passed a verdict of guilty despite an appeal to the judge for clemency” means that clemency was probably requested before the verdict was passed (it could have been requested by the defendant’s lawyers, the jury, or a third party) and yet the jury STILL gave a guilty verdict, something that does not show leniency or mercy. “Despite” signals that even though something was in place or asked for, it didn’t change the course of action, which in this case is a guilty verdict.

Hope that helps!