i would develop the sites for different purposes with different content or redirect the lesser-or-undeveloped domain to the more established domain.
So no issues of having 2 sites indexed with the same info?
No problem however one should rank significantly better than the other and that one may suffer a little dilution owing to the duplicate content.
Certainly they won't get banned since there is no such thing as duplicate content banning however site 2's SERPs ranking will be much, much lower, probably at the end of the rankings therefore ensure you launch the site you want to be seen in the SERPs first and then add the othe site later on.
If I were you I would develop the secondary site as a mini site with different but relevant content rather than a duplicate but with direct links to the main site, even use the same header, footer and navigation menu if you like but just give it a few pages of different content.
a domain redirect means only one site is going to get indexed.
Heavens, it was far too early in the day for me to be answering! :-)
phranque is correct, a redirect automatically points at the domain you wish to be indexed, there are no duplicate issues however what you can do, if your hosting service provides it, you can redirect and leave BusinessName.com in the url however I wouldn't recommend that since that is where you can get duplication issues.
Gimme fresh coffee:-)