The main difference food-wise is that certain foods are not kosher for Passover, and these vary depending on whether you are Ashkenazi, Sephardic, or Mizrahi. Leavened breads are not kosher for passover, and have to be disposed of before passover (there's a ritual cleaning out of the pantry). In some traditions, you also can't eat rice, corn, or other grains.
There are different dishes that are traditional, but of course this would vary by tradition. In my family, chicken would be more for Shabbat, but for passover we'd have brisket, tsimmis, kugel, and a few more things.
There are foods for passover that are symbolic, but not so much for Shabbat dinners. For passover, there are hard-boiled eggs, parsley, matzoh (the unleavened bread), horseradish, and charoset (which is sometimes made with apples, nuts and wine, sometimes with dates, and sometimes with other ingredients).
Also, for Shabbat dinner, the ritual aspects are mostly confined to prayers over the bread and wine, dipping challah in salt, ritual handwashing, and a prayer at the end of the meal, and sometimes good fun singing songs. Passover seder rituals can last for hours, and involve lots of prayers, 4 glasses of wine, lots of handwashing, the retelling of the story of Exodus, and weloming the prophet Elijah. So it's a much more involved evening. I absolutely love it, but am glad not to do it every week!