Impossible to know the situation without more information but in general, you need to 301 redirect (preferably page by page) from the old site to the new site. Otherwise, all value in search terms will be lost, as will links/bookmarks to the old site.
If that's not practical at the present host but you are using your own domain. Use reliable low cost/free hosting, point the domain and add a .htaccess file. That's all you will need, no requirement to move the site as the site files are not accessed.
Assuming the new host has standard database facilities, possibly a one click install of phpBB, no real problem to just move the existing board:
Beyond that, will help if you provide the address of the current site.