The other answer is a pretty good list of differences, so I'm going to try to complement it with a fluffier, more hands-on answer that I think better describes the experience of riding with hub gears. I switched from derailleur to a Shimano 8-speed hub two years ago. My daily commute was 8 km each way back then, in all weathers.
The short answer is that you don't have to think so much. I often call hub gears "Homer Simpson gears": gear goes up, gear goes down, gear goes up, gear goes down. It's that simple.
With derailleur gears, while coasting on the approach to a junction, I would do one revolution of the pedals and change down, just in case I would have to stop at the junction. If I didn't stop, I'd have to change back up as I started pedalling out of the junction. Now I don't bother with that. If I know I have to stop (e.g. when coming up to a red light), I can change down while coasting; if I have to stop unexpectedly (e.g. at a pedestrian crossing, or a give-way with poor visibility), I can change down while I'm stationary. I never have to start in the wrong gear. It took me a month or two to get out of the habit of doing that one rev while changing down.
I did an organised charity ride last year. I could always tell when I was coming up to a hill, by the crunching noise of people leaving it too late to change down, and I sailed past them effortlessly. Hubs share the derailleur problem that they can't change gear while there's too much force on the pedal, but when changing gear on a derailleur you have to wait for the chain to completely locate on the new chainring, whereas it takes less than half a second for the hub to click into its new gear. This makes it much easier to ease off the pedal a little to make the change, even on the hill, and means you don't need to plan ahead for it.
People are often worried about not having so many gears on a hub as they did with derailleurs. I've found the range on mine pretty adequate. The bottom "granny" gear is great for getting stuck behind a family going up a narrow bridge. Once or twice while going downhill I've wished for an extra gear or two at the top, but I have to be really putting my foot down on the flat to even get into 7th. Mainly, of the eight gears I have, I use 4, 5, and 6 on the flat. I couldn't imagine finding the range inadequate unless you were often hauling a heavy trailer or live in a mountain range.
There's one difference bigger than these day-to-day things. As I say, I've had this bike for two years. It lives outside, and gets ridden in all weathers. It did about 5000 km in its first year; less in the second year as my commute is shorter now. It hasn't been back to the shop for a service yet, and I'm still on the original chain (which I clean and lube less often than I ought to). It still rides as smoothly today as the day I rode it home from the shop. In fact, it's smoother now, as the 4→5 change was a bit sticky for the first month. (Apparently you often get some slightly sharp edges when they're new, and it just takes a little use to smooth them off a bit.) The efficiency is still pretty good, though I'm sure the chain has stretched a bit, and the gear changes are perfect. I don't see the behaviour you get from derailleurs, where as the chain wears, the top and bottom gears become harder to get to. I don't have to keep adjusting the tension of the shifter cable, and the chain never comes off. And according to the man in the shop, he won't need to service the hub for another year yet. Add this to the reduced chain wear you get from never pulling the chain at an angle, and you get one reliable ride.
- less shifting
- less thinking when you do shift
- no getting stuck in the wrong gear, and you can drop whatever "coping strategies" you used to avoid that
- less maintenance (but one big maintenance job every few years)
- less flexibility for extreme loads and inclines
- more consistent performance over long distances