What are some real life examples to understand the key role of assertions?
To recap (and this is true of many languages not just Java):
"assert" is primarily used as a debugging aid by software developers during the debugging process. Assert-messages should never appear. Many languages provide a compile-time option that will cause all "asserts" to be ignored, for use in generating "production" code.
"exceptions" are a handy way to handle all kinds of error conditions, whether or not they represent logic errors, because, if you run into an error-condition such that you cannot continue, you can simply "throw them up into the air," from wherever you are, expecting someone else out there to be ready to "catch" them. Control is transferred in one step, straight from the code that threw the exception, straight to the catcher's mitt. (And the catcher can see the complete backtrace of calls that had taken place.)
Furthermore, callers of that subroutine don't have to check to see if the subroutine succeeded: "if we're here now, it must have succeeded, because otherwise it would have thrown an exception and we wouldn't be here now!" This simple strategy makes code-design and debugging much, much easier.
Exceptions conveniently allow fatal-error conditions to be what they are: "exceptions to the rule." And, for them to be handled by a code-path that is also "an exception to the rule ... "fly ball!"