I liken IGH vs Derailleur as automatic vs manual:
the inside of an IGH is planet and sun gears, pawls and clutches, bathed in oil, not unlike an automatic transmission.
Derailleurs are harder to use because you have to be pedaling and plan ahead what gear you want to be in (especially when nearing a stop).
IGH: select a gear and that's it
IGH is a lot a lot easier to clean and less likely to be damaged. With an IGH you can install a full chain case and basically have nothing exposed to the elements in your drive train (think dutch city bikes, which even have roller or coaster brakes to complete the low-maintenance / sealed away design)
Derailleurs are more efficient and lighter (like a manual, If you know what you are doing)
However, here is where IGH really starts to break down:
derailleurs are a lot easier to work on, easy enough to be done at home with simple tools. For DIY Mustachian types, this is very appealing.
But if you DONT want to work on your bike, any bike shop in the world can fix a derailleur problem, and for very little.
IGH are much more expensive to start with
IGH do in fact wear out, and when they do often the repair is not worth it (only some bike shops CAN work on them). You can wear out an IGH faster by shifting while pedaling
When your IGH does fail or have problems, not only is it more $ to fix but consider the hub is part of your wheel, so new IGH = rebuild the wheel = more $$$; Fixing or replacing a derailleur or cassette does not involve rebuilding your wheel. (perhaps in Europe where IGH are a lot more popular this is less a problem, but in the USA derailleurs rule supreme)
IGH requires horizontal dropouts (because you must use the position of the wheel to tension the chain) which means no quick releases, which means getting your rear tire off and back on a PITA to fix a flat. Derailleurs can use simple vertical dropouts with QR so popping the tire off and on to fix a flat is easy.
You can take one look at a derailleur and determine its condition. An IGH can hide serious problems
Personally, if I was to use an IGH it would be because I was building the ultimate "dont have to think about it" town bike, think 10-12mph / 10-12 miles max per day
1) The most basic of 3 speed (pick a chainring that puts the gears where you want them for your terrain)
2) full chain case
3) either the most basic of rim brakes, or a coaster back / roller front
4) tannus airless tires (yes, they are slow, and rough, but remember we are talking 10mph / 10 miles max or some sort of bombproof tire (Schwalbe marathon plus) and or tire armor (tannus armor)
5) heavy steel step through frame with rear rack, panniers
6) upright swept back handlebars
7) flat pedals.
Basically, an omafiets!