I covered some of the uses in the article, but there are a slew more I didn't.
The main reason I use routers on business trips is because someone's with me (this used to happen a lot more). In a scenario like that, a router is imperative, because it allows both people to share the same Internet connection. There's also the networking factor, of being able to shift files from PC to PC without having to touch the other PC. In the case of this particular router, because you can connect to a wireless access point with ease, you could even enhance the signal strength, which is something I'm looking forward to trying out.
For someone who's alone on a business trip, the biggest use is the ability to untether yourself from the desk, if there's a wired connection there only. Most desks I've had to deal with at hotels generally aren't that comfortable, especially for hours at a time, so being able to kick back in the much more comfortable chair is definitely preferred.
I'm sure there are even more scenarios I'm not thinking of, but these are the ones I've taken advantage of many times in the past.