What is the difference between going to the court date on your summons and requesting a trial for a photo radar ticket?

Whether you hire a lawyer or not is a personal decision. To the extent one can afford to hire counsel, most lawyers will tell you that you should not go to court without one, which is my view as well. Also, there are certain defenses or motions that typically are more successful with a lawyer, especially where there is a question as to who the driver was, as in your case. As far as the presented options on the ticket, either way you are entitled to a hearing/trial. You can show up on the date set to appear and then set it for trial or ask for a continuance, or you can ask for a hearing via the ticket and you will or should be notified of the hearing/trial date. There usually is no motion for a dismissal "before" the hearing based on a lack of evidence or inability of the city to prove you were the driver; that is why they have hearings/trials. But on the other hand sometimes you (or your attorney) can convince the city to dismiss the case if there is an evidence problem. This is an example of where having an attorney improves the chances of success, not only from the perspective of generally persuading the prosecutor to dismiss, but it is more difficult, and in some cases impossible, for the driver to say "that's not me" etc.