Some html elements have JS events that behave differently when true/false is returned.  For instance:
<input type='submit' value='Click Me' onSubmit='ValidateForm();'>
...vs...
<input type='submit' value='Click Me' onSubmit='return ValidateForm();'>
In the second instance, if the ValidateForm function returned false the form will not submit, in the first even if the function returns false the form will still submit.
I think this scenario, you can see the different between using the return keyword and not.
UPDATED To simplify, if you use the return keyword you are passing a value back to the function that called the onsubmit.  Without it, you are simply calling the function that you name in the event handler and do not return anything.
UPDATE 2021-01-21 This functionality also work for the onclick method on html anchors / links (a):
Sample Usage:
<a href="#never-used" onclick="alert('click clack'); return false;" > 
Click Me