Each has its own advantages, in my mind. Both are equally effective referents as both invoke the ritual of eating with others, "to dine" and, "to sup." I find that I like the sound of "taking sup" over "dining," perhaps because it appeals to my ego to use slightly archaic words and usages. On the other hand, I am unaware of the existence of the descriptive "supper plate."
There are disadvantages to each, as well. "Dinner" carries a connotation of a more formal process or a higher-quality meal, while "supper" is taken as a simple, "common" meal. I would also admit when considering the homophones to preferring to deal with a noisy din as opposed to something that could be described as "suppurating."
The debate rages on, a tempest around tea time.