Sync and Publish Addons

Does “Publish” addon include the “Sync” addon?
If I want to have both sync and “Publish notes to the web” do I have to pay for both addons (4+8 USD)?

If these are both separate, how does it work when I want publish a note to the web?
It’s stored on Obisidian server, but I can’t edit it from another device or…

No, they’re separate. Sync is for keeping your vault updated across your devices, Publish is just for sending chosen notes to a public site.

Publish doesn’t sync your vault. You can publish from any device, but you still need some way to get the vault onto that device.