I've experienced this myself -- when a company I worked for briefly split in two and became one company with two websites. It was a difficult decision, and even more difficult transition.
There are the obvious complications that come with re-branding a business: new logos, new colors, new business cards, new letterheads, etc. These are the same problems transitioning companies have dealt with for years. But today there is so much more to worry about.
Today, rebranding businesses have to consider the transitions in the digital world as well.
There's website domains. Is your URL available? Or how much is it going to cost you to purchase it from that squatter?
There are SEO effects. Sure 401 redirects will combat a complete domain swap. But all the work you've done creating inbound links from other highly-ranked websites; all the research on META tags, keywords, and descriptions; and any positive effects you've received from social search (SPYW) is as worthless as having a Facebook page with zero "likes."
Which leads me to the biggest issue -- and the one that eventually put an end to Netflix's rebranding fiasco -- is the question of what to do if your handle isn't available on the social networks?
Just as there are domain squatters for URLs, there are Twitter and Facebook handle squatters. As soon as it becomes apparent any social network is the "next big thing" and opens itself up to business and companies, users sign up and sit on social network names. Heck, I'm not even squatting and I have 27 Twitter accounts and 13 Facebook usernames. But unlike URLs which have democratic companies monitoring and maintaing these domains (i.e. GoDaddy), social network handles are a ruleless wild west for name-squatting cowboys (and cowgirls).
The moment Netflix announced they were rebranding their DVD mailing service as Qwikster people jumped on net and blew up search engines with the term "Qwikster" -- which inevitably lead them to find a foul-mouthed, pot-smoking Elmo who already owned the Twitter account for @Qwikster. It was major flaw in C-suite's plan for Netflix. Something you'd expect a company as successful as Netflix to have thought of. But they didn't. And they soon lost the quickdraw gun fight with Jason Castillo (aka @Qwikster) who's since cleaned up is act a bit -- changing his profile picture to a crest and washing his mouth out with soap -- but not before single-handedly putting an end (and some bad PR) to the Netflix / Qwikster debacle. And...