Live Book not accessible after change of internet provider (yet dashboard is ok!)

Since it works with the old one vs the new router. Compare the configs (page by page) between them.

There has been cases when FW software running on the desktop/laptop caused the issue by blocking the ports. but in this case the only change has been the router. Unless there were other changes as well.

it seems you keep the same IP when connecting to both routers. Are you set to static? That should not cause any issues though.