I notice that it is considered up to code to install a GFCI in the absence of a proper ground, but that it must be labeled (link). Why is this? Does the GFCI have some drawbacks compared to a proper ground wire? Is it safe for people but dangerous for equipment?
GFCI is actually the ideal solution for the people problem (shocking people) and serves that purpose better than grounding. Assuming no boneheaded bootlegging of grounds. But GFCI (alone) does nothing for the ESD/lightning problem which can damage sensitive electronics or cause goofy problems with radio transmitters.
Current travels in loops. Current wants to return to source. For natural electricity like ESD and lightning, the source is ground, not neutral.
You need fair warning of the lack of equipment ground, hence the labels. The labels also allow you to pass inspection on 3-prong outlets without a ground.
You've gotten some good answers, but I thought I'd add a few thoughts.
1) A ground serves as a low-resistance path away from the load so that you or some object doesn't serve that purpose instead. I saw a demo where a teacher put his hand in a 5-gallon bucket of a grounded and submerged running drill. He said he had nothing to worry about because electricity preferred the copper ground to him and his boots. (Haven't tested this personally; never will. I have put my fingers in a water bath with my thumb and middle finger perpendicular and parallel to current flow and only the parallel position makes your nerves all tingly because of potential!)
2) A fuse or general circuit breaker is slow and detects an overage in current or short-circuit, and disconnects the circuit from the service to prevent overheating, arcing, and fires. It works too slowly to prevent shock and save lives.
3) GFCI has different electronics inside which senses a current imbalance very quickly and can save lives, particularly if they don't have a delay built-in. That's why they're required outdoors where you and damp ground can serve as a path to ground.
4) AFCI which is the most expensive and most recent addition as a circuit breaker goes beyond GFCI because it has electronics that can compute whether or not a circuit is experiencing atypical arcing that wouldn't trip either of the previous types thus preventing fires. It's also, as you can imagine, the most expensive.
Practical difference? I've tripped quite a few residential circuits (I often work on them live because I'm stupid and lazy) and the difference between using cable cutters on a standard breaker and a GFCI is that you don't get sparks and a big, melty hole in your cutters on the GFCI which doesn't require you to go to the panel and throw the breaker. Instead, you get a quick pop at the receptacle which can reset on the spot.
Remember that when properly configured, GFCI's only protect "downstream" from the device, and if not grounded, can be misleading since they the third prong may create the appearance of an active ground (like in this house I recently moved into where I went a year before pulling it and finding there was neither a ground nor a metal box and conduit!)