I went over to the local Apple store the other night with my iPad 2 in hand, downloaded the Omni apps, and did some experimentation to answer this very question for myself.
My conclusion was that the only noticeable performance difference as far as running the Omni apps went was that I didn't run out of RAM as quickly. The things that take a while with my iPad 2 took just as long on the iPad++ within the accuracy of my measurement technique. I was surprised enough by this that I went and checked model numbers just to make sure I wasn't using a demo iPad 2 :-) The time savings for me would come from not relaunching apps, not speedier execution. Do enough switching between apps and that might be better than faster execution but more relaunching. The display, on the other hand, offered a substantial improvement for reading text. If looking to buy an iPad as a dedicated producer or consumer of text, I would go for the new one, no question about it.
Haven't managed to convince myself that we really need to be a 3 iPad family, though, and was able to walk away without too much regret. If I had bought a 16 GB WiFi-only model last time around, it would be a bit easier to "justify" the updated model, but a 32 GB LTE model (to keep the same level) is almost half again as much. Might have to start doing something more lucrative than answering forum posts :-)