European RCD (differential) protection, at least in countries using Terra-Neutral (TN) earthing as North America does, is focused primarily on protection from fire, with a secondary helping of shock protection under reasonable conditions. The theory behind this is based on catching dielectric breakdown and associated leakage before it becomes a full-fledged carbonized (tracked) fault that gets hot enough to cause surrounding materials to smoulder, and it's generally accepted as a sound one.
However, in order to provide cost-effective protection for an entire distribution board, a small number of RCDs are used as feeder devices for a larger number of breakers. This is practical because of the small number of circuits required by a typical flat or even a house under IEC-based systems and due to the lower amperages required as a result of all appliances being 230V at a minimum. However, the combination of higher native leakage due to the higher operating voltage and that multi-circuit span forces the use of a higher IΔn setting, typically 30 to 100mA in modern practice, than would be ideal for shock protection.
The North American GFCI approach, however, is focused primarily on shock protection, and in particular, protection from secondary effects of shock on skeletal muscles in addition to keeping people from being killed by the shock disrupting heart rhythms (ventricular fibrillation). These secondary effects include both involuntary skeletal contractions ("can't let go") and the more insidiously lethal effect known as electric shock drowning, or ESD for short. This is where a mostly-submerged individual (nominally a swimmer) loses muscle functionality due to low-level (tens of mA) AC currents flowing in freshwater and drowns as a result, and is reflected in the original NEC GFCI requirement being not to protect bathrooms, kitchens, or any other place we think of as requiring protection, but swimming pool equipment such as lighting. (Nowadays, low-voltage transformers are used for this instead.)
The upshot is that North American GFCIs are sensitive enough to provide excellent protection from shock; however, to achieve that 5mA sensitivity, they require a bit more circuit complexity than a RCD, which leaves them more vulnerable to failures over time and also means they can't be reverse-fed without causing permanent damage. They also are more sensitive to nuisance tripping due to cable leakage than a 30mA RCD, which means that placing them in the panel increases the nuisance tripping risk vs. the receptacle (socket-outlet) type of GFCI most commonly seen in North America.
Furthermore, the reason we don't put them everywhere is because some appliances (still) aren't compatible with GFCI protection, or would be counterproductive to protect with a GFCI. For instance, the NEC itself prohibits GFCI protection on circuits serving fire alarm control panels, as you don't want a ground fault somewhere to disable the primary AC supply to the alarm panel for an extended period of time, run the batteries dry, and thus cause the fire alarm to not work when a fire starts. Likewise, refrigerators are generally not GFCI protected due to issues with hermetic compressor insulation breakdown causing nuisance tripping combined with the risk of food spoilage involved.
"Why didn't the US adopt the IEC approach?"
If you are wondering why North America didn't keep the sensitive GFCI protection against shock, yet adopt higher-level RCD protection for catching house fires, the reason has to do with the difference in voltages between the two systems. In a 120/240V system, lights and most small appliances run on 120V while only heavy appliances run on 240V. This has the upside of limiting the voltage to ground while making two utilization voltages available, but has two downsides. First off, the lower voltages are less stressful on wire insulation, but the higher currents used increase the risk of a glowing fault at a bad connection causing a fire, which is much harder to catch than insulation breakdown leading to a cable fault.
Second, and more importantly, the higher currents involved mean that North American main breakers (incomers) need to be much larger and feed more branch circuits to feed an equivalent level of electrical utilization. (The US NEC requires a minimum of 60A service for most things, and 100A for single-family houses, whereas a 100A service would be considered quite large by European standards as I understand them.) This increases the cost of breakers considerably, and in particular, since North Americans use circuit breaker mains (incomers) instead of switch-disconnectors, having to incorporate RCD protection into them would drive costs up significantly and create space issues within already cramped panels as well. (The US already had "Rule of Six" split bus panels when services started exceeding the 60-100A range, and the current generations of main circuit breakers used are quite cost and space-optimized compared to their molded-case equivalents.)
There are places where North America did learn a thing or two from international experience, though. One of these was the protection of electrical trace heating cables used for freeze prevention in various places; due to the high potential for damage to a heat cable placed outdoors somewhere, the NEC requires what is known as ground fault protection of equipment, or GFPE for short, which is essentially a 30mA or sometimes 100mA IΔn RCBO (as they only come in breaker form) implemented using the same parts as a breaker-type (RCBO form factor) GFCI. The NEC also calls for ground fault protection on high-power three phase services (1200A and up) due to the high degree of fire risk posed by faults on such systems.
Enter the AFCI, stage right
Continued concern about electrical fires in North America led the CPSC, UL, and electrical distribution manufacturers to examine the IEC approach, with sensitive instantaneous breaker trips and RCD protection. However, there were issues with this; most importantly, 120V motors generate too much inrush current for IEC B and C curve breakers, with their sensitive magnetic trips, to handle. As a result, North American MCBs use a less sensitive magnetic trip, closer to the IEC D curve, which renders them relatively insensitive to arcing short circuits (vs. the much less common bolted fault).
Nonetheless, the North American distribution equipment manufacturers pressed forward with creating a device that could try to catch arcing short circuits, insulation tracking breakdown, and other such pesky firestarting phenomena. A couple different approaches were tried: Eaton/Cutler-Hammer came up with a "branch/feeder" AFCI design that combined GFPE with analog detection logic, while other manufacturers looked to a microprocessor based signature recognition approach. Fractures during the standard-setting process led to compromises, though, and a GFPE requirement was not incorporated into the final AFCI standards; instead, a different requirement for "series arc detection" made it in, leading to the creation of combination and outlet branch circuit AFCIs.
Furthermore, early AFCIs were fraught with installation issues; their GFPE protection made them sensitive to improper neutral connections and required the use of multipole AFCIs when multiwire branch circuits were encountered, while signature detection proved to be a finicky technology to refine due to the sheer variety of sources and signatures out there, as well as the need to be robust against "normal" arcs caused by switching and motor commutation and a wide variety of RFI sources. The former issue led some manufacturers, most notably GE and Siemens, to drop the GFPE function from their AFCIs altogether, while the latter has been a constant (albeit slowly improving) thorn in the side of AFCI technology, leading to quite a few callbacks for electricians even as the NEC AFCI mandates expanded quite considerably.
However, the NEC has also incorporated the requirement for 30mA ground fault protection for marinas due to issues with leaky boat electrical systems causing electric shock drowning hazards in lakes and rivers. Will this lead to a meeting of the two sides of the debate? Will AFCI technology come of age after an extremely long and painful childhood? Will arc-fault detection fare better in the rest of the world than it does in North America? Only time will tell...