What is the difference between sanity and smoke testing?

Smoke tests refer orignally to hardware testing, wherein a device would be plugged in and if it started to smoke then it was unplugged as no further testing was required. It doesn't work, there is no point in testing it.

Sanity tests are, traditionally (in other realms of testing) tests that establish whether or not a result could be true, rather than if it is true, that is to say if the output is "sane". If you put "9" into the function "squareroot()" and get "badger" you have failed a sanity test. If you get, say, a positive number less than 9 then you have passed it, even if the number is not "3". This is why they refer to tests that ensure that the system is working enough to be tested - it is giving a sane response to basic tests, therefore it is not a waste of time to run full functional tests.