Education in the U.S. varies so much by state and by local community that it's hard to generalize, but to me they are synonyms. In my town, the elementary schools actually do have "elementary" in their officials name; that is to say, "Barton Elementary School," "Coke Elementary School," etc. I don't think the US/UK distinction Sarah mentions is perfectly clear. For example, I see that a U.S. government website, https://www.usa.gov/school , says Primary and Secondary Education The Education System in the United States follows a pattern similar to other countries: Early childhood (or pre-kindergarten) education Primary (or elementary) school Middle (or junior high) school Secondary (or high) school Postsecondary (college, career, or technical schools) education

Dan is right. Both are used universally here. However it can refer to grades k and 1-3, or 1-4, and after that there is another level where I live called Intermediate School then Middle School/Junior High, then High School

They're the same, but "elementary school" is generally preferred in the US, and "primary school" in the UK.