Assertion:

In Assertion if the verification fails in any of the testcase then the further execution of same testcase will not take place and the error will be logged but then the next testcase will be executed.
For eg: If we are on the login page and we have to verify that after successful login we should navigate to home page. In this scenario, we will put assert in our testcase and the further steps will not be executed if after login user did not land on home page.

Verify:

In verification, if the verification fails then that failure is logged but the execution remains continue. For eg If we have to verify the title of any button and on failure it will not stop executing the remaining steps.