The Knox case highlights a few of the differences between the Italian appeal process and the American apellate system.
First, in Italy, the parties are allowed to present new evidence to the court on appeal. In Amanda Knox's case, experts testified about flaws in the DNA evidence. In America, an appeal strictly reviews what happened in the trial court. No new evidence is permitted. If new evidence in the case is uncovered, it may be brought in a writ of habeas corpus.
Second, the defendants in an Italian appeal are permitted to make statements. Knox proclaimed her innocence and pled with the jurors for her freedom today. Such statements are unheard of in an American appeal, or even at trial.
On appeal in America, the attorneys for each side present legal arguments primarily in written briefs. The attorneys may then give oral arguments to the court, but even that is sometimes optional. Just as there is no new evidence permitted, no witnesses or statements may be heard. Even in an American trial, a defendant may not make a statement unless he or she testifies, as any other witness, and submits to cross-examination by the other side.
Finally, an Italian appeal is decided by eight "jurors." The jurors consist of six members of the public and two judges, including the judge who presided over the appeal. In America, an appeal is decided solely by judges. There are no jurors involved. There is usually a panel of three judges in the intermediate appellate court. There may be as many as seven or nine judges in the highest court. The California Supreme Court has seven justices, for example, and the United States Supreme Court has nine.
In short, an Italian appeal is more like a second trial, whereas an American appeal is a strict review of the original trial.