if you're using "Run whether user is logged in or not", then you should NOT tick the "Do not store password" checkbox. Otherwise, how is it going to use your account to access network resources, without using the password?
Also, try changing the setting to "Run only when user is logged in" and just set a time trigger to make the schedule task run by itself. See if that works or not.
Lastly, try running a very simple Powershell script instead of the one you're trying to run. In your test Powershell script, just do something like:
write-host "test"
and just add a -noexit to your scheduled task arguments (under Actions). What should happen is that your schedule task will run and will leave the Powershell window open, which then allows you to verify whether the script has output the word "test" or not. Try this with both "Run only when user is logged in" and "Run whether user is logged in or not".
If both are working fine, then the issue is not with scheduled task and it'll be an issue with your actual script.