The short version: Google Sites is a good option if your school already uses G Suite for Education and you don't want to introduce a new set of usernames and passwords for people to have to remember.
- Easy to embed files from Google Drive (Docs, Sheets, Slides, Forms).
- Easy to embed calendars and videos.
- Easy to invite other teachers and or students to collaborate on site development.
- "New" version of Google Sites is optimized for mobile display.
- URLs assigned to Google Sites are long, cumbersome, and nearly impossible to remember. Don't believe, try to get all of your 7th graders to this site in ten minutes or less https://sites.google.com/a/freetech4teachers.com/civics-with-mr-byrne/ and then try to get them to remember it.
- While design options have definitely improved in the new version of Google Sites, they're still far behind what you'll find on Weebly.
- Support for embedding content from providers outside of the Google ecosystem has improved, but is still lagging behind other website creation services.
- No support for a blog section within the new version of Google Sites.
The short version: Weebly offers a Weebly for Education product which is free and is preferable to the standard Weebly free product because the education version doesn't display advertising. If you're not invested in G Suite, then Weebly is a slightly better choice.
- Weebly for Education lets you manage up to forty student accounts in one free teacher account.
- Large gallery of design templates that you can customize to your liking.
- Includes option to have a blog section within your site.
- Supports embedding content from many 3rd party sources.
- So many options that it can be a bit overwhelming to first-time users.
- Annoying pop-up message trying to sell you a custom domain appears every time you publish a new page.
- Doesn't have a collaboration option to let you invite other teachers work on a site with you without also giving them administrative rights.