I think it depends on the type of business you're operating. In many -- most -- cases, the brand
is basically the product and arbitrarily separating them out is only going to add to confusion and extra work.
In other cases you may have multiple, "first class" products that are independent of each other and your identity as the provider of those products. I'd definitely separate it then.
Unasked, but I think relevant: you should absolutely separate out your infrastructure from both your product and business. To carry your analogy forward, there would be facebook.net, fb.net, or perhaps even some code word thats used to refer to internal resources (servers, internal tools and resources, etc). It's fine if they're indirectly exposed to end-users or easily discovered.
