If you'll be using only the new domain and not the old, you can just use a server side 301 redirect with .htaccess from the old to the new. Don't have the same content show up with both domain names, that can be a problem.
Keep the old at least until all the incoming links are changed from the old to the new. And if the old is a company name that's known, or that people might type in, you might want to hold on to it so it doesn't fall into the wrong hands.
Duplicate content is never good. So if you want to change your domain name use the new domain and use a 301 redirect on the old one. The will keep all of the traffic from the old site and not agrivate any spidered engines. 301 means moved permanently so spidered engines will spider the new content and disregard the old, there by no duplicate content.
When I recently had a client change their domain I then went through all non spidered engines (yahoo, dmoz, etc) and submitted the proper requests to get the domain name switched in all of the listings. Now it is finished and they didn't lose any traffic. I just did it one engine at a time and was very careful.
<added>When someone goes to www.olddomain.com it will send a 301 header and redirect them to www.newdomain.com since that will now be the active site.
We thought if we had redirection fees, we may as well chuck in some extra and have two ranking sites to be indexed....
my personal opinion - yes
and I do have experience with promoting similar sites from the same company/person
I have seen them
get delisted because they had the same phone number
delisted because the whois resolved to the same company/person
refused listings in directories because of similar content
had both listing well and then have them both get buried because of similar content
gotten pr0 for similar content
really isn't much fun, I have had clients make second sites themselves with out telling me. I have had my boss setup mirrors of our own domain with out telling me (we are still banned in google 1yr+), I have had people come to me asking to fix the fact that these scenarios had happened previously, I have even done it myself for a trio of sites and spent more time running around being really careful trying to prolong the inevitable and insulated the main site so the fallout would be minimal. I had the client sign an agreement that he accepted these risks before we went into it.
If you try to promote two sites make sure the content is 100% different and even go so far as to get another phone number and make sure you reference nothing from the other site. But be fully aware that if you get pegged you could lose them both. remember that in directories you are, more than likely, going to end up in the sam category or one very closely related.
So, my recommendation is to build one good site, use the method I referenced above and only promote one. Use the time and money you would spend to produce the second weaker version of your site and put it into link campaigns and other programs that can help bolster your primary site and not dilute your efforts by trying to focus on multiple sites and possibly lose them all.
all imho and experience.