Even with all the same parts, the laptop has the laptop form factor, which means it’s trades off cooling (peak/sustained performance) for mobility and cordless operation. A desktop with its large coolers will just have more mass and bigger fans - it’s going to run cooler and quieter. It’s just physics. At the same time, being able to fold up a laptop and bring it with you where you go is super convenient and obviously worth the trade off for many users. And we’re at the point where performance required for most people is far exceeded by current CPUs (notwithstanding companies trying to keep costs as low as possible and trying to cut things down the the line of acceptability in a race to the bottom).

I’m interested in bigger picture conversations. Germanium has a lower melting point but is more conductive than Silicon. Would it make an ideal material for a mobile processor? Using less power while not having the expectation of pushing the limits of heat?

A desktop is more flexible, can be upgraded, expanded, can run cooler, and is easier and less expensive to repair. If you don’t need portability tower is the obvious choice.

Power consumption is highly variable in both classes, while laptops generally consume less power it is because of component selection for the purpose of battery life.

The same components and settings in a tower would consume less power as there is no load from the battery. You get some of this power back when you use the battery but not as much power as you put in. There are losses both in and out, expressed as heat.