There is indeed basically no maintenance for a heat pump, other than cleaning the filter. Just as an example, when was the last time you maintenanced your car's air conditioner? That's also a heat pump, and in those you basically just clean the evaporator filter and, if it leaks, get the leak fixed and fill it back up with refrigerant. If the fan or thermostat fail, you'll know it, because it will stop working.
Modern and highly efficient heat pumps have variable speed pumps in them and are designed to run 24/7 rather than cycling on and off, because it's more efficient.
I suspect your sensation of comfort has most to do with relative humidity and the constant movement of air (rather than short bursts of much warmer air), but I'd like to hear others' opinions.
As for cost, heat pumps can vary drastically in efficiency. Central units tend to be less efficient than mini-splits, presumably due to the large overhead of large fans and the heat loss through ducting. New units can be as much as 2-3x more efficient than older ones, and a lot of improvements have come in the last 5-10 years.
I have a spreadsheet that calculates the cost of various heating methods based on local prices. We have high-ish electricity costs (around 16 cents per kwh) and it tends to be that a very good heat pump is (based on average efficiency) marginally more expensive than natural gas and around the same as wood. Oil is around 50% more expensive, propane costs around 2.5-3x as much to heat with, and electric resistive is a little more than propane.
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Anecdotally:
When we bought our mini-split last year it was the most efficient one on the market. It's responsible for the heating of half of our house, with a pellet stove covering the other half. Our home is in Vermont and around 1250sq ft, and built in the 60's without heavy emphasis on insulation.
During the transitional seasons (when it's most efficient) we didn't even notice a change in our electricity bill - keeping the house 10-30F warmer than outside was $x per month at most, and we didn't fire up the pellet stove unless it dropped below 40 outside.
In winter, our electrical bill has increased $80-100, with outside temperatures typically being between 10-30F and with some days being as cold as -10F. The pellet stove by comparison burns $100-120 in pellets during the coldest month. The side of the house with the heat pump is considerably more pleasant than the one with the pellet stove - the pellet stove is not controlled by a thermostat but cranks out pretty consistent heat, is noisy, requires the addition of fuel every day, and needs half an hour of cleaning soot and ash weekly. Its fan circulates a lot more air in the room which can feel a bit chilling to exposed skin. If we planned to stay here forever I would not hesitate to put a second mini-split in and reserve the pellet stove only for those weeks below maybe 10F.