Well, the answer is pretty simple. isset($var) returns whether or not a variable exists and is not null, where !$var tells you if that variable is true, or anything that evaluates to true (such as a non-empty string). This is summarized in the first table of this documentation page.
Also, using !$var will output a notice that you're using an undefined variable, whereas isset($var) won't do that.
Mind you, they are two different things:
<?php
var_dump( isset($foo) ); // false.
var_dump( !$foo ); // true, but with a warning.
$foo = false;
var_dump( isset($foo) ); // true
var_dump( !$foo ); // true.
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