The difference between smoke testing and sanity testing:

Smoke testing – This type of testing is done initially to determine if the new software is crashing the systems in every 5 minutes or corrupting databases, the software may not be in a 'sound’ condition to proceed for further testing in its current state.

Sanity testing - Testing to determine if a new software version is performing well enough to accept it for a major testing effort. If application is crashing for initial use then system is not stable enough for further testing and build or application is assigned to fix.

There is a small difference in sanity testing and smoke testing. In smoke testing,only major functionalities will be tested in sanity testing, all functionalities will be tested. both are done before UAT.
Smoke testing is after build releasing we will test high level functionalities working or not. Sanity testing is we will test all functionalities working properly or not.