What each term usually means
Temp mail (temporary email) emphasizes the short lifetime: an address that exists for minutes and then disappears, ideal for a single verification code. The focus is on it being disposable and automatic.
Burner email borrows from "burner phone": an address you deliberately keep at arm's length from your identity, used once for a signup you do not want traced back to your main inbox. The focus is on separation from your real account.
Where they overlap
In practice both are a random inbox with no signup that you throw away after use. On this page the address above serves either purpose: paste it into a form, read the message here, and it self-destructs after five minutes.
Neither is a private mailbox. Both are public while live and readable by anyone who knows the address, so the "burner" separation is about keeping mail off your real inbox, not about strong anonymity.
Which to use
If you just need a quick verification code, think of it as temp mail and move on. If your goal is to keep a specific signup off your primary address, think of it as a burner — same inbox, different reason.
For anything you must log into again later, use neither: a disposable address is gone in five minutes and cannot be recovered. Keep it to one-off gates and trials.