Reading some of the stuff on here I'm beginning to suspect it's a technical nightmare to get it set up right. This seems to go against the grain of cycling as a simple, low-tech sport so I'll stick with cables.
Naaah, it's a doddle to set up. Admittedly you have read about some snags, particularly with more exotic setups like mine, but that's present in mechanical systems too. The average set-up is a non-issue. It works out of the box.
The sophistication of the system is hidden by its simplicity. It works on a CAN bus (Controller Access Network). This reduces the number of wires required since a single cable link can go from component to component. The wire is only 3mm thick and very flexible. It can be made invisible on your bike, especially if you have a Di2 frame or, you modify your frame with a port or two.
How do you set it up? You bolt on all your components as usual, install a battery either in the seatpost or at the water bottle and, connect the two, three or four wires. It works immediately and out of the box. Only when you want to change the standard functions, do you have to bother with software. For instance, if you want to swap which buttons do what or, want to try (you don't have to) some sort of fancy intelligent shifting.
Even then it is easy. It can connect to your smartphone via Bluetooth LE (Low Energy) and from there you can do some stuff - not endless nonsense, but useful things. Shimano's software is bug-free and super well thought out.
The system can integrate with ANT+ devices so that you can have all your data and readings on one device. If you don't want comms, you can buy a cheaper junction box.
Mountain bikers can go further and connect their compatible suspension to the system and "tune" it, if tuning is your thing.
Road bikers can add extra shift switches wherever they want them. On the top of the bars, on the bar ends - the choice is endless.
Your bike will require charging two, maybe three times a season. Charging cycles are quoted in kilometers (2000 is typical) but that is of course nonsense. If you change gears a million times, then battery life will be shorter. When you theoretically run out of power on a ride, the system shuts down non-essential functions first, before stranding you in a single gear. Stage racers don't have to charge between stages. Compare that to stage races where you have to change cables, particularly on MTB races. I have done such races where a cable service was required every single night. I could have slept during those hours instead of fiddling with dirty cables.
To me, the biggest advantage is getting rid of gear shifter cables and cleaning up the frame. Second advantage is zero adjustment. Adjusting cable derailers isn't a big deal if you know how, but zero adjusting is just better.
Lastly, it comes in mechanical and hydraulic brake version. This means you can be a retro grouch and be electronic or, go full hog and get with the program, disc brakes and all.
Bring me that fancy new Spez of yours and I'll drill it for Di2.