your desktop computer will be faster and likely have better graphics.  that will let it support larger monitor at higher resolution or more frequent refresh.

Neither term has a set definition.  You would need to look at the specs of a specific laptop and a specific PC to see what the difference is between them.

You are correct that main advantage to a laptop is that it is portable.  While with a desktop PC you have more space in the case so you can upgrade it or add more devices (such as a second hard drive.)  With a laptop you are pretty much stuck with whatever comes build in to the laptop.  The desktop also normally gives you more ports and more options for adding external devices such as controller.  You get a longer lifespan out of a desktop because you can upgrade components rather then have to replace the entire unit.  

You have to decide whether portability is more important that upgradability.  then make your choice accordingly.

All gaming computers/laptops produce heat. A gaming computer has more space inside it, so the design can afford to have more powerful components as heat dissipation is less of an issue.

In gaming laptops the chips have to be optimised for power vs amount of energy/heat produced so there is heavy customisation/design of parts e.g. cooling, motherboard design so the laptop won't overheat. In laptops the only parts that can easily be bought off the shelf are RAM. Everything else is customised for that series of laptops and therefore expensive

In PCs a lot of parts are generic and can be easily bought  off the shelf, relatively cheaply as a lot of manufacturers will be producing the same thing. This means the costs are lower.

Additionally with PCs the end user can service some of the parts - the CPU,RAM the graphics card can easily be replaced, the cooling system can be upgraded etc. With a gaming laptop you may be able to upgrade the RAM but that is about it.

Basically you will get more bang for your buck with gaming desktops vs laptops