To att to DrChaos's reply, you can in effect have different stylesheets so long as all users use the "usedefault" stylesheet. You must also have identical stylesheets on both forums to avoid that "breaking up" of the forum.
Redirecting one domain is one way of doing it and you should have a look if you will require a second license with that option.
If you do have two installations for two databases then you will definitely require two licenses and it is irrelevant where your database is. For all you care it can be on a third domain, since .threads connects with the host, username and password. So long as a particular host - where each domain and threads installation is - with a username and password is allowed to access this database you will be ok.
Regarding banning you are right since banning is done on a database level. Regarding filters you will have to do it on both installations since it is file based and will exist on one installation and not on the other.
Regarding installation of modifications it has to be done on both sites. Regarding rotating banners yes you can have different ones.
Search engines will query each site as a separate board because it is a separate domain. I don't know if you will be penalized for duplicate pages but I don't think that it will be the case.
