They are doing different things beyond just the “normal” timed 100% power option. Every manufacturer is different of course, but generally “potato” lowers the power for an extended period of time, “pizza” might kick on a top heating element and increase the fan, etc. Many “popcorn” settings are actually quite sophisticated nowadays, and can listen to the audio signal of the pops to determine when to finish a bag of popcorn. Technology Connections actually has [a great video about this](https://youtu.be/Limpr1L8Pss). When in doubt, the manual for your specific microwave usually explains the different settings in detail.
The microwave can control time AND power levels, just like controlling cooking time AND the amount of flame under the pot on a stove top. For a soup on the stove for example you may want to have the full flame in order to bring the water to boiling, but then minimum flame to just keep it simmering and/or not burn the bottom layers.
With microwaves the heat is applied to the water IN the food, and a potato vs. a pizza have different distributions of where the water is. So the microwave may heat the topping of the pizza and then reduce power to let that heat penetrate into the dough, and may on and off the power levels to heat the (outer) potato layers and let that heat penetrate inward.
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