Smoke Testing Vs Sanity Testing

1) Smoke Testing is also called as BVT-Build verification testing or Tester acceptance testing or TESTABILITY Testing.
2) It is the first Testing (MANDATORY on every s/w build) we do on every software build release.
3) Here, Test lead or testers will share few fixed set of main functionalities of overall build and execute them,once thy are passed then Test lead gives/send mail to dev team that smoke/BVT is passed and we can approach to further real Testing(other test cases).
4) If the (the mail functionalities are not working) then it will be rejected by QA and no further testing is done. In this case - the dev team will fix issues and again release/deploy the build.
In short - When developers release s/w build for Testing, Testing team can download or launch build into 1cabin system and then Testing team can execute fixed set of cases related to main functionalities of that s/w build to confirm that s/w build is working or not.

SANITY Testing:

1) After release of new s/w build with modifications or fixing of raised defects - we do retesting of those defects (Say module 1).
Once retesting is done, the other test cases of same module should pass with out any side effects/new defects  is called SANITY Testing.

2) Now, we should also test or execute other modules(2, 3...or test cases which are related to module1 in module 2 or module 3...) So that the fixing(code changing) doesn't effect the other modules also is called Regression Testing.