Anything that uses power gets hot because there's always some loss and the loss is converted into heat. But heating is not necessarily the same as overheating.
Laptops and desktops and their power supplies are designed for continuous use without overheating, but anything in heavy use can break down after some time. There's no black-and-white answer and we have to balance our needs with the risks.
You could gain some additional safety margin by blowing on the laptop and its adaptor with a small table fan. Or, (and this is something I've thought of just now) you could place the laptop AC adaptor on top of an up-ended flat-bottomed aluminium pot!
The aluminium will conduct part of the heat away and dissipate it more easily than the adaptor alone.
This is like comparing apples and oranges, as the saying goes. A desktop
may be more suitable for continuous use, but your primary concern is power saving, in which case the laptop wins.
All the usual things, really. Processor, RAM, HDD, graphics, etc. This will help evaluate the power consumption.
One possibility that may be considered by those addicted to downloading large files all the time is to have a dedicated downloading machine. Downloading files is a non-demanding task. A d/l machine could be built up of old discarded but working parts. A low power CPU like an old Celeron or Sempron, 128MB RAM, etc could be used. The low power used by such a configuration could be reduced further by underclocking.
For example, I still have a 500 MHz Pentium III CPU and a working motherboard with 128MB SDRAM. If I were a heavy downloader, I could use that purely as a downloading machine and it would be just as good as a modern machine for that purpose.