How is the riding experience of a Rohloff hub different than derailleur gears?
I can only compare the basic Shimano Acera and Deore (3x8) deraileurs to the Rohloff 14-speed hub, but these are my observations.

On the Rohloff hub


Rohloff, like most hub gears is slightly less efficient than a well-maintained derailleur, as this PDF study explains. In my experience the difference is not noticeable, perhaps because I've mostly used cheap derailleurs.
Changing gears is always impossible if significant force is on the drivechain.
A short pedalling pause is required to shift gears.
It is possible to change to/from any gear when stationary or coasting.
Gear inches change between gears is simple and predictable (roughly the same percentage per click).
The gap between the gears is fairly large.
Some gears are noisy (eg 7th)
There are a couple of gears which need a bit of care shifting or you end up temporarily in top gear.


On a derailer


Most gears run quickly and efficiently. Several gears are noisy/inefficient due to chain angle.
Changing gears is sometimes tricky if there a large force on the drive chain. It depends on which change is being made, some are really easy and smooth, others need some care.
No pause to fast, gentle pedalling is required to shift gears.
Only possible to change gears when pedalling.
Two shifters means that shifting quickly to a predictable-length gear a skill that requires a bit of thought.


YMMV, but I find the Rohloff to be efficient and simple to use. We use it on a tandem, so it may be that the pause needed when shifting is less distruptive on a solo bike.

However, I also expect that an experienced racer would be able to shift their high-spec gears much more quickly and efficiently than any hub gears and that this might mean that derailleurs are the only option for racing. That and the fact that you can't seem to connect them to brifters.

(reworked from this answer)

There are a number of (mostly positive) opinions from Rohloff owners here.