Here is a simple and understandable difference that can clear your confusion between smoke testing and sanity testing.

SMOKE TESTING:

• Smoke testing originated in the hardware testing practice of turning on a new piece of hardware for the first time and considering it a success if it does not catch fire and smoke. In software industry, smoke testing is a shallow and wide approach whereby all areas of the application without getting into too deep, is tested.
• A smoke test is scripted, either using a written set of tests or an automated test
• A Smoke test is designed to touch every part of the application in a cursory way. It's shallow and wide.
• Smoke testing is conducted to ensure whether the most crucial functions of a program are working, but not bothering with finer details. (Such as build verification).
• Smoke testing is normal health check up to a build of an application before taking it to testing in depth.

SANITY TESTING:

• A sanity test is a narrow regression test that focuses on one or a few areas of functionality. Sanity testing is usually narrow and deep.
• A sanity test is usually unscripted.
• A Sanity test is used to determine a small section of the application is still working after a minor change.
• 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.
• Sanity testing is to verify whether requirements are met or not, checking all features breadth-first.