The browser extension is a lightweight and blazing fast alternative to our desktop apps, but it only works within your browser and has some known limitations (for instance, it may not work as expected with Google Docs, while the desktop app works flawlessly wherever you use it).
In short, you should go for the browser extension if you only plan to use your canned responses in the browser and don't want to install an app on your computer, and the native apps if you want a fully-fledged solution to manage and use your canned responses in the browser and in other apps (Slack, Word etc).
In the current state of the art, we do not recommend installing both as the quick shortcut commands may interfere with one another (the extension and the desktop app will try to autocomplete your typing at the same time).