School supplies would be anything that you are required to bring to school, but it usually refers to stationery. School things could be any things which are related to school, but this often refers to your school bag (and it's contents) and uniform.

In addition to the the other answers, I would say these two phrases mean basically the same thing but maybe would be used in different contexts. For example, we often say we're going shopping for school supplies. But school things might be used in the context of a mom telling her child, "go get your school things, it's time to go." Hope that's helpful!

In my country, I have never heard anyone say school things. We say school supplies commonly. It can mean anything needed in a classroom: pens, paper, erasers, pencils, markers, etc. Example: The teacher has to buy all her own school supplies out of pocket. Mary forgot her list for this year's required school supplies. Tom bought all of his school supplies already.

"School supplies" is the usual expression and includes pens, paper, textbooks and things that students need to help them with their studying. "School things" is not an expression I have ever heard in use, but presumably it means "school supplies". It could mean a lot more depending on the context. For example, a school flag or T-shirt is a "school thing" but these aren't the sort of items that students need to help with their studies, which is the usual connotation of "school supplies". So they are alike, but "school things" is much more general, I would say.