they are very similar words. brief means quick or short (as in time or length)
concise means brief, but also complete and/or informative. it is associated with information and communication
i associate the word “concise” with efficiency and purpose, whereas “brief” doesn’t necessarily have intention behind it. for example, a brief essay is probably short because the writer got a little lazy and wanted to stop writing, while a concise essay is likely edited and intentionally short. but that doesn’t mean that “brief” is always negative— you might send a brief letter to a friend, for example, to stay in touch with them, or you might eat a brief meal so that you aren’t late for work.
also, “brief” is a more fluid word, and you can use it to describe a wider variety of things, while concise is more typically associated with writing and communication
for example:
a brief meeting
a brief meal
a brief walk (you would not use concise for any of these)
versus
a concise letter, a brief letter
a concise speech, a brief speech
a concise argument, a brief argument