No, they are not synonyms, although they are related. A cliché is a phrase that is repeated too often and has become boring and unoriginal from overuse. Here are some examples of stale old clichés (at least, they are cliché's in the US): "We need to think outside the box." "[Name of country] is a land of contrasts." "[Something] represents a paradigm shift." "Corny" is actually a reference to the areas of the United States where corn is grown; that is to say, areas far from the city where the people live in small towns and are not socially sophisticated. It basically means "unsophisticated." It has two main uses. One is to describe jokes that are not very clever, or are so old that everybody already has heard them. "My uncle is a really nice guy, but he tells a lot of corny jokes." Here's a corny joke: Q: "Why did the chicken cross the road?" A: "To get to the other side." "Corny" can also mean overly sweet and sentimental. "My wife really likes the 'Mitford' books by Jan Karon, about very sweet people in a lovely town being nice to each other, but they were too corny for me."

This is correct. Corny is much more informal. Cliched is a more formal way of saying the same thing.