"Can they use different stylesheets?"
*** If a user logs in under domain 1 and chooses a style and then goes to domain 2, he should still have the same style because its using the same database for information. Just make sure that both threads databases have the same stylesheets. If you have crimson style on one and not on the other, when the user logs in they will get a plain white stylesheet and start asking why the forums are messed up
"In principle it will be two different sites but the content and users will be the same on both?"
*** Yes, 2 different domain names, the same content on both domains.
"How about the PM's?
Will a PM sent on one board show on both boards? and when deleted it will get deleted from both boards?"
*** Yes, still using the same database.
"Will the burden on the server duplicate or will it remain pretty much the same?"
Yes, because both domain names will be using the same database so the load will be no more then normal.
This also has to be on the same server and each with its own threads license.
Would it be easier to add a re-direct under one domain name to point to the other.
I own a few domain names that all point to my site. I bought haloce.org (currently number 1 in google searches for haloce

) and it forwards to my leetgamers site.