The simple answer is preamp distortion results when you distort/overload a guitar amp’s preamp, and power amp distortion results when you distort/overload the power amp section. If you have a master volume amp, when you turn the gain up on your amp, but keep the master volume low, you are likely getting preamp distortion. Preamp distortion generally offers higher gain than power amp distortion and is more compressed and smoother, with higher sustain at lower volume levels. Preamp circuits are where the amp’s EQ controls live, and amplifiers that provide preamp distortion generally offer lots of gain shaping possibilities.
If you keep your preamp clean, but turn up your amp’s output/master volume until it distorts, or if your amp has no master volume control and you turn it up until the amp begins to distort, you’re hearing power amp distortion. In general, power-amp distortion is considered to be richer, more dynamic, punchier, and less compressed. However, depending on the wattage of your amp, achieving distortion may require ear-splitting volume levels. And power amp sections rarely have much in the way of tone shaping, so you don’t have the breadth of EQ options.
In general, vintage-style amps sound best when driven into power amp distortion — depending on the amp, its preamp may not even be able to distort. Modern, higher gain amps are built to do most of their tone shaping in the preamp, and run relatively clean power amps, though you can often turn the amp up to either achieve pure power amp distortion (with a clean preamp setting) or a combination of preamp and power amp distortion.
As a rule, if you want to experiment with power-amp distortion, your best bet is to buy a lower-wattage tube amp, in which the power tubes will start “cooking” at a lower volume.