What you call the evening meal says where you are from
Both words are correct, and both have meant the evening meal for a long time. But they did not start out as synonyms. For centuries, dinner was the main meal of the day, eaten around midday, and supper was the lighter meal eaten in the evening. In much of rural America, the South and the Midwest, that older system is still alive: dinner at noon, supper at night. In most cities and suburbs, it has been replaced by lunch and dinner.
So the vote is not really "which word is right." It is which daily rhythm you grew up with.
Two French words for two meals
Both words came into English from French. Dinner comes from Old French disner, the meal that broke the night's fast — a cousin of the modern French déjeuner, and the same idea as English breakfast. Supper comes from Old French soper, linked to soupe: the evening meal was often a bowl of soup or broth with bread.
For most of history, people who worked the land ate their biggest meal in the middle of the day, when they needed the energy, and something simple at sundown. The names followed the schedule.
How the cities moved dinner
The shift began among urban elites and spread with industrial work. As people commuted to offices and factories, the midday meal shrank into something quick, eaten away from home, and the main meal moved to the evening, when the family was back together. That evening main meal kept the prestigious name, dinner, and a new word, lunch, filled the gap at noon.
Farming communities had no reason to change, because their day had not changed. That is why the old pattern survives in agricultural regions, and why so many Americans still hear "supper" as the word of their grandparents' kitchen table.
The holidays that keep the old meaning
Traces of the older system are everywhere once you look. Thanksgiving dinner is often served in the early afternoon, exactly when dinner used to be. Church suppers and the Last Supper both point to an evening meal. Sunday dinner, in many families, still means a big meal after church rather than anything at night.
Some people also use the two words to mark a difference in tone: dinner for a meal with guests, supper for the everyday family meal at home. The distinction is not in any rulebook, but it is real for those who keep it.
Vote for what you call the evening meal on an ordinary weeknight at home. The tally will show how much of the old schedule has survived.
Sources: etymologies of "dinner" (Old French disner) and "supper" (Old French soper) per standard etymological dictionaries; Merriam-Webster entries for both words; social histories of American mealtimes.