Sanity testing  
Typically an initial testing effort to determine if a new 
software version is performing well enough to accept it for 
a major testing effort.  
 
Smoke Test”, a broad but shallow verification of all the 
critical functionality

Smoke testing is non-exhaustive software testing, 
ascertaining that the most crucial functions of a program 
work, but not bothering with finer details. 
 
Sanity testing is a cursory testing; it is performed 
whenever a cursory testing is sufficient to prove the 
application is functioning according to specifications. 
This level of testing is a subset of regression testing. It 
normally includes a set of core tests of basic GUI 
functionality to demonstrate connectivity to the database, 
application servers, printers, etc.

smoke means,checking whether all functionalities present in 
application or not.it is done by developer.Before going to 
give the build to testers,developers will do this.
where as sanity means checking whether all main 
functionalities are working fine or not.it is done by 
testers.
we won't write test cases for smoke testing.

Smoke testing is normal health check up to a build of an 
application before taking it to testing in depth.

Where as 

Sanity testing is to verify whether requirements are met or 
not,checking all features breadth-first.