What do Verify Disk and Repair Disk actually do in Disk Utility?
Disk Utility in many respect is powered by fsck, a command line utility that deals with file systems (not just the one found on OS X).

The option to verify (and repair) your disk deals with the health of the file system (HFS+) and the integrity of the data that resides on it, rather than the physical disk. It looks at how the data is stored and not if the drive is reporting mechanical errors (to some extent this is reported by the S.M.A.R.T status). However it should stand to reason that a hardware fault will likely permeate through the file system structure (though not always).

Is Disk Utility good at finding hardware issues? No, not really. It's meant more to isolate problems with your data set than it is your hardware. There of course is some continuity there but they are not monogamous.

DiskWarrior and Drive Genius will undoubtedly do a much better job diagnosing the physical health of your drive. But none are as powerful as the utilities released by your respective hard drive manufacturer. If you can use those, then all the better. Using multiple tools may also prove to be an affective strategy. 

Another avenue which can help in discovery is to comb your Console's logs for anomalies. I've had a WD disk exhibit strange behaviors and first became cognizant of the problem by way of a single error during the initial boot up sequence. Google and WD's native Windows diagnostic tools helped me isolate the issue (to some degree).