I don't know that summary notes and prose summary are technical terms, but if I were to take a guess, I would assume that one is less formal than the other.
As a teacher, if I gave an assignment to write a "note" of summary, or to take "summary notes," I would accept bullet points, lists, and incomplete sentences as correct. Generally, the objective of such an assignment would be to make sure students had the basic understanding of the plot line or details of a text.
If I gave the direction to write a "prose summary" on the other hand, it is likely that I am looking for a couple of things. First, of course I want to see that my students understood what they read and included necessary details in the summary. Additionally, prose suggests that this summary needs to be written in complete sentences, and likely these sentences need to form complete paragraphs. Not only am I assessing basic understanding of textual details, but I'm looking for technical correctness and writing skills as well.
A "prose summary" is a short statement that expresses the main ideas of a passage. These statements provide an overview of the major ideas in a reading passage and leave out minor ideas or details. Summary notes, on the other hand, involve boiling down the text into note form. For example, if you are taking notes on the reasons the Allies won World War II, you would place that topic at the top of the paper (or computer document) and then, off to the right, include each reason (in as succinct a form as possible). Under each reason, you would place short summary details. Taking summary notes allows you to remember what you read because you reduce the reading to its essential details, and by writing you transfer information from your short-term to long-term memory.
A prose summary is when you asses the grammar, diction, and tone used in the piece read to give a better explanation of the theme.