What is the advantage of installing a desktop client for 1PW over just using "1Password" extension?

You can also manage items en masse in the desktop app - for example, copying a bunch of items from one vault to another.

(For anyone who is interested, you can select multiple items in 1Password for Windows by using ctrl + click to select multiple individual items, or ctrl + a to select all items in a list. Very useful if you have a bunch of items in one vault that you decide you want to delete, archive, or move to another vault without having to click, move, click, move for each one!)

Another advantage, though not used by most, is the desktop client let's you drag passwords from the desktop client into non-browser interfaces, like SSH, Toad, or RDP sessions. Some of those mediums (RDP in particular) often don't support copy & paste so that's a huge benefit for some.

Hi amswitzer:

When using the desktop application of 1Password, in addition to being able to easily manage and edit your items from an application, instead of needing to use my.1Password.com, you also gain the ability to keep 1Password in your browser unlocked for however long the 1Password desktop application is unlocked. Additionally, because the 1Password desktop application is handling the unlocking, Windows Hello can be used to unlock 1Password for Windows, and by extension, unlock 1Password in your browser as well.

Jack