Well, if you are trying them out tomorrow, why not try out both? Its important to remember that there is no "better" in this case, only different. Both basses have a devoted following that swear by their choice and shun the other. Most here can appreciate the beauty of either one and that its strictly a matter od preference.
Here are a few things to look for....
The Jazz has a thinner neck, the Precision neck is fatter. Most people have a preference one way of the other in this area, so play both and see how each neck feels in your hands. Remember, thinner is not automatically better. Some folks find that a thin neck is fatiguing to their hands, others can't stand how the P-bass neck is. Only you know which is better for you.
Jazz basses have a more cutting high-midrange tone, Precision have thick low mids for a deeper, fatter voice. Listen to them and see which one sounds better to your ears.
Try to play them both through the same amp and set the EQ on that amp flat. That will give you a better idea of each basses true voice. One should definitely speak to you more than the other, when that happens you will have your answer. Good luck and update us with your decision after you go.