Three-phase power is really about delivering more power with less copper by 1) sharing the neutral conductor (or getting rid of it) and 2) using lower currents which can make do with much thinner conductors.
The difference between one three-phase feeder and three completely separate two-wire circuits is that the three phases, which are 120° apart, share a single neutral conductor. In fact, if the load is perfectly balanced across the phases and currents are sine waves, they will cancel each other out, so neutral will carry no current at all and can be omitted! But in practice, there still usually is a neutral conductor to carry the imbalance (except for specific loads like delta and star motors under certain conditions). In total, we only need 4 wires instead of 6 to carry the same amount of power. That's 1/3 less copper! (Excluding PE of course.)
Now, compared to using a single two-wire circuit rated for three times the current, three-phase also wins by a long shot. (We're again assuming a balanced load and equal power factors.) This is because resistive heat scales with the current squared, so 3 times the current produces 9 times the amount of heat at the same resistance. So, we need to use thicker conductors to lower the resistance. But the area of its cross-section grows quicker than the surface area to dissipate heat, so the maximum current a conductor can safely carry scales sublinearly with its cross-sectional area. For example, 1.5mm² will suffice under most conditions to carry 16A, but for 48A we'll typically need 10mm².
All in all, the rule of thumb is that three-phase AC gets you the same amount of power for roughly 25% less copper compared to single-phase AC!
Some loads also make use of the rotating magnetic field, like motors (see below).
Lastly, lower currents will reduce heating in case of loose connections, reducing the risk of fire. On the other hand, a 400V RMS phase-to-phase shock is a lot more severe than a 230V RMS phase-to-neutral (or phase-to-PE) shock.
Common use cases (load-side):
- Motors. Three phases produce a rotating magnetic field, perfect for simple self-starting motors. (Arguably, this is less relevant today with more modern motor designs, though three-phase induction motors are still widely used.)
- Electric cookers (usually split by cooking zones and oven, like one phase for two zones each, and the third phase for the oven)
- Water heaters (especially those that heat water instantaneously on demand)
- EV charging (even an ancient European three-phase installation is likely able to deliver 11kW of power and can therefore fully charge an EV overnight)