In Test Mode the burner performs everything but turning on the laser, so the disc isn't actually burned. It's useful for any situation where you want to simulate a burn but don't want to waste a disc, e.g. to test if your system can burn DVDs at 24x. Notice, though, that a successful test doesn't mean the real burn will be successful since a lot of things can change while doing a real burn.
As for the Verify, it compares data burned on the disc against the source image/files, and if you get read errors/miscompares then you know it's a bad burn.