For the two power measurements, one is instantaneous power, the other is average power. Different situations call for different measurements.
Your grid power consumption is probably measured in terms of current alone. That means that the number on your power bill is different from the amount of energy your devices dissipate as heat. That measurement is converted into power using the RMS trick, but there is no guarantee that it will exactly match the actual power consumption of your house.
Your UPSs has different limitations for real power, current, and peak current. And your house wiring (or better, any wiring) has limitations for (windowed) average current (with many different limits for different window sizes). Wiring also has a limitation for voltage, but it's usually high enough to not matter on normal usage.
As a rule, with AC, there are so many degrees of freedom that any question becomes hard to answer.
Reporting power is easy with the power triangle. It's a mathematically exact concept.
Measuring power is a different story. Distortions of the assumed perfect sine wave create measurement errors. Meter bandwidth matters, but most have low bandwidth.
Grid meters at least use voltage phase, if not magnitude. Residential customers are billed only for real power, so the phase information is necessary to exclude the reactive power from the measurement.
One rating is for maximum output. The other describes a short-circuit that you can create with an overly reactive load.
The degrees of freedom complicate the implementation, but the bottom-line discussion abstracts away those details.
Being out-of-phase is like turning a crank with one hand while resisting with the other - wasted effort. Some of the VA becomes VAR instead of W.
The math thing is a simplification so you can take measurements with a multimeter, and do just a bit of arithmetic to get all three power values. It comes from the average power integral of trig functions with complex numbers.
Watt rating is what you'd expect. Voltage and current are working together to do work. The limit is probably thermal.
VA rating at max current is a short-circuit. If your load draws enough reactive power, the voltage and current get so far out-of-phase that a huge current sneaks through without much voltage. It could be many times as much current as the max current at max Watts.
According to Wikipedia,
> Some devices, including uninterruptible power supplies (UPSs), have ratings both for maximum volt-amperes and maximum watts. The VA rating is limited by the maximum permissible current, and the watt rating by the power-handling capacity of the device.
Is there something besides the "maximum permissible current" that influences the "power-handling capacity" in a circuit? These sound the same.