Smoke tests are tests which aim is to check if everything was build correctly. I mean here integration, connections. So you check from technically point of view if you can make wider tests. You have to execute some test cases and check if the results are positive.
Sanity tests in general have the same aim - check if we can make further test. But in sanity test you focus on business value so you execute some test cases but you check the logic.
In general people say smoke tests for both above because they are executed in the same time (sanity after smoke tests) and their aim is similar.