They all use the same tool under the hood (Lighthouse). Currently the version on web.dev doesn’t let you change the configuration settings, it runs with the same default settings as the extension and the devtools version. So, if you’re ok with the defaults then they should all give similar results.
One nice bit about using the web version is if you sign-in and add a URL, it will save your scores over time. When you run the audits again at a future date we’ll show you how you improved or areas where you maybe regressed. This can be helpful for identifying if and when you may have shipped something that slowed down performance or one of the other categories.
We have a lot of plans to continue building on the sign-in experience to make it more useful for folks over time :)