While not under the "corporate umbrella", don't think that they will be completely independent. They will have the same owner and will have to coordinate route structures to avoid direct competition and thus eroding both carriers profits. It's all about maximizing profits in Dubai and they will do what it takes to accomplish that. Years ago, they actually had something called a yield management committee or something like that made up of the airlines serving Dubai. In reality, it was a price fixing cartel - something that would be illegal in most places in the world.
Over time I would expect to see a migration of lower yield, short to medium range flying going to the newer carrier, allowing Emirates to concentrate on higher yield, long range operations. It might even be an attempt to address the service issue. Right now, Emirates has to serve a broad range of "markets" with different service requirements. "Unloading" some of the low yield markets on this new carrier might make management of the service delivery a little easier.
Overall, I think that this is probably a shrewd move on the part of the Dubai government. One carrier will specialize in the low yield, low cost markets thereby probably generating more profit from them while the other is then freed to specialize on higher yield, longer haul markets. If the route structures are coordinated, overall profit is maximized. It's very telling that they have chosen Emirates head of commercial planning to be the head of the new airline (while not relinquishing his old post).