First, let me just correct your second sentence:
I love fish and vegetables, which are the most common foods in Japan.
Fish and vegetables are plural, so the rest of the sentence has to agree. Now, the concept to understand is that "which" is non-restrictive, whereas "that" is restrictive. Have a look at these:
I'm interested in attending one of the Ivy League schools, which are among the highest-rated schools in the world.
I'm interested in attending one of the Ivy League schools that are among the highest-rated schools in the world.
The first one says two things: I want to go to an Ivy League school. Ivy League schools are among the highest rated in the world. The first sentence doesn't restrict the first one; it provides additional related information.
The second one says only one thing: I want to go to an Ivy League school, but only an Ivy League school that is among the highest rated in the world (by implication, not one of the other Ivy League schools that is not among the highest rated in the world). This sentence is restrictive; the "that" clause restricts or qualifies the main clause.
Here is an article that covers the basics of the subject and gives more examples.