Just to confirm which one you are getting prompted for and having an issue with, as there are several passwords you can set up:

One is a local Security password for launching POL itself (configured off the Security Settings button, BEFORE you click on a user ID), which is saved locally.

Then there is your POL password for game login (legacy from the old system) that is also saved locally. This also has to be saved in the SE Account Management system--so if you change it on your PC, you have to change it at the website too. This is setup on the Settings button after you click on a User ID. You will also be saving your SE ID on that settings page. If these settings are not in synch with what is saved at the website, you will not get in.

Then there is the SE password (this password does not save) and the token password if you are using it. These passwords are setup only at the website and not saved locally--you have to enter them every time you log in to the game.