In the United States, "diploma" means the actual physical piece of paper, the certificate. The "degree" is the intangible status you have. For example, I have the degree of "bachelor of science" from MIT. My diploma is the piece of paper that says so. It's signed by the president of the university. I could lose my diploma--the piece of paper--but I will always have my degree.
In the United States, you graduate from "high school" at about age 18. "High school" is more formally called "secondary school." At the graduation ceremony, you are given a diploma. So you have "a high school diploma." But for some reason, you do not get a "degree" from high school. The word "degree" is used for completing "post-secondary" courses of study. An "associate degree" usually requires only two years of post-secondary, so it is not as high a degree as "a four-year college degree."
"Diploma" is derived from Greek roots meaning "folded in two." Centuries ago, diplomas were made of parchment, a very-high-quality paper-like material made from sheepskin, and they were typically folded in two. But diplomas have not really been made of sheepskin since the 1800s.