The main advantages of hub gears over a 1xN derailleur system are reliability and being able to shift while stationary. Since there's no derailleur or chain tensioner it's easier to add a full chaincase which means chains last a lot longer and with mudguards added there are no dirty bits accessible on the bike. They're heavier and harder to maintain if you have to maintain them.
There's a whole style of bike that uses those advantages (and others) and from a US perspective is called a "european" or "dutch" bike. Broadly, it's a bike built for someone who is not a cyclist, rather they are someone who uses a bike for transport. It should just work, no special clothes or rituals required.
Compared to a 3xN derailleur system a hub gear is much simpler to operate and maintain. All the gears are in order and you only have one shifter. The compromise is a smaller gear range (Rohloff being the obvious exception), ranging from a little smaller for the Shimano Alfine 11 down to "yes, there are two gears" for the back-pedal-shift 2 speed hubs. But in return you get extremely long life for the chain and cogs (10,000km or more for the chain, 20,000km+ for cogs) and a general low-fuss maintenance requirement. That also makes it practical to put a hub gear in places where derailleur gears could go but you wouldn't want them - under my load carrying quad, for instance, where the derailleur hung down between the rear wheels just waiting for a rock to break it off, and turning the thing over for maintenance is a pain.
Removing the rear wheel on a hub geared bike ranges from nigh on impossible (some of the Shimano hubs, and some dutch bikes have chain cases that are just irritating to work on) to almost trivial (a Rohloff with QR). The problem is the shifter cable - you have to detach it to remove the wheel. Good ones just unhook, poor ones you have to undo the adjuster barrel and re-adjust the gears on reassembly. Shimano are getting better, but Rohloff and Sturmey Archer have it down pat.
Servicing hubs is not often done these days. The manufacturers generally allow bike shops to swap out the internals for new, saving the hassle of building a new wheel, but that's about it. I had three Shimano Nexus 8 hubs fail at 5,000km intervals, and every time I just got replacement internals rather than the hub being repaired (the last time, of course, Shimano had stopped making the hub so I had to throw away the wheel). With Rohloff the same thing might happen, except after 150,000km or more... most people don't ride that far in a lifetime. Also, with Rohloff you change the oil every 5,000-10,000km, at a cost of half an hour of your time and about $20 for an oil change kit (or $10 for just the oil).
I usually run a Schwalbe Marathon Plus on the rear of my hub geared bikes, which trades a little extra rolling resistance for huge puncture resistance. It's part of the "my bike just works" philosophy. I'd rather reliably take an extra 30s to get to work than at random intervals be 5-10 minutes late and dirty due to a puncture (because you know that you'll only get punctures when you're in a hurry and it's raining).