The Ashkenazic rabbis (probably around 10th or 11th Century AD) in Germany also prohibited what they called kitniot -- basically, anything that was sold in grain-like form or might be confused with grain. Their prohibition included rice, millet, corn, and legumes. When in doubt, prohibit, on the grounds that it is better to be more strict in avoidance of foods than to be less strict and potentially violate the Biblical commandment not to eat leaven. Of course, corn (maize) was not known to Europe at the time, but when the New World crop made its appearance a few centuries later, it was quickly identified as kitniot and thus prohibited for Passover.
Sephardic Jews (from Spain and the Mediterranean areas) did not prohibit rice, millet, corn or legumes. They said there was no danger of confusing them with prohibited grains. So a cheesecake made for a Sephardic family might use cornstarch. I discussed this with a friend who is a Mexican Jew of Ashkenazic heritage; he said it was always difficult at Passover, his Sephardic friends could eat corn tortillas and he could only eat cardboard. (That's an in-joke reference to matzah, unleavened bread.)
Over a thousand years or more, there have been some other local variations, with local customs arose forbidding various other foods on Passover. Sugar is an interesting example of this. In ancient codes, sugar is forbidden on Passover because sugar was often adulterated with flour, which was cheaper. When sugar came in solid cone shapes, and today when sugar is processed by machine, the fear of such adulteration is eliminated and sugar was declared usable for Passover. Another local custom: some families do not use garlic on Passover, but no reason is known.
Potatoes, on the other hand, were never prohibited by anybody ... and hence Passover baking tends to potato starch.
And, it is worth noting, the prohibition against cornstarch doesn't apply to the rest of the year, only to Passover. The cornstarch doesn't make food "unkosher," it makes it hametz -- "leavened," unfit for consumption on Passover.