Smoke and sanity testing are both performed after a software build to identify whether to start testing. Sanity may or may not be executed after smoke testing. They can be executed separately or at the same time - sanity being immediately after smoke.
Because sanity testing is more in-depth and takes more time, in most cases it is well worth to be automated.
Smoke testing usually takes no longer than 5-30 minutes for execution. It is more general: it checks a small number of core functionalities of the whole system, in order to verify that the stability of the software is good enough for further testing and that there are no issues, blocking the run of the planned test cases.
Sanity testing is more detailed than smoke and may take from 15 minutes up to a whole day, depending on the scale of the new build. It is a more specialized type of acceptance testing, performed after progression or re-testing. It checks the core features of certain new functionalities and/or bug fixes together with some closely related to them features, in order to verify that they are functioning as to the required operational logic, before regression testing can be executed at a larger scale.