Will I have to re-embed a form after applying changes at Form Builder?

If I make changes to my form and save those changes, do I need to publish it with a new URL?  Or once I've made the changes and saved them, the revised form will automatically appear on the published website?

Your second suggestion is correct. If the form is embedded, changes applied at Form Builder will be reflected automatically on your website. Form's URL link will also remain unchanged. 

However, there is one exception. If you use the Source Code to embed a form, then you will have to re-embed it with updated version.

We will gladly assist if you need more help.

Thank you.