The difference is in how you approach the substitution:
On a fixed price menu it is very common for restaurants to not substitute, even Disney. However, if a person has a special dietary need, that's different. For example, say someone is ordering from fixed price but cannot eat potatoes due to an allergy or diabetes . . .
If there is that type of special need, from what I've learned here, the best thing to do if you have special dietary needs is to call ahead and let Disney know. You can do this by calling the dining line or by calling the restaurant directly.
If unable to contact the restaurant in advance, then when you arrive, ask to speak to the chef personally to let him/her know of any special dietary needs, and then ask if the dish you're craving can be changed to suit your needs.
Disney is pretty wonderful at doing this!
However, just asking to substitute a veggie for potato when the menu specifically states "no substitutions" . . . well, that may not work.
That said, I agree that the waitress could have been a whole lot nicer about it . . . or maybe offered suggestions or SOMETHING. No excuse for poor service.