Well, except that's not what's going on. Since your outer quotes are double quotes, you'll get variable interpolation regardless of the fact that single quotes appear inside them. You can test that with this:
$foo = 'hi';
That will output 'hi' (with the quotes). There actually is no difference between your original query and the solution posted. This is easily tested as well:
$date2 = 'second date';
$time2 = 'second time';
$phonecheck = '555-1212';
$query1 = "UPDATE WRGleads SET date2 = '".$date2."', time2 = '".$time2."' WHERE phone = '".$phonecheck."'";
$query2 = "UPDATE WRGleads SET date2 = '$date2', time2 = '$time2' WHERE phone = '$phonecheck'";
echo $query1 . "<br />\n";
echo $query2 . "<br />\n";
You will get two identical lines of output. If one worked and the other didn't, it's because of some third factor not in evidence.
Sorry to, I'm sure, add to the confusion, but it's not going to help you any to go through life with mistaken ideas about string quoting because of this.