Answer:
Well, as it is a gaming device and you are not travelling from one LAN party to another, you should get a desktop PC.
Advantages:
- it is cheaper (more power for a lower budget)
- it can be improved very easily (In case you want more ram or a better graphic card, just buy the stuff and exchange it with the old components)
- the screen is usually bigger and you can have any screen you want (you can surely connect your laptop to a screen, but what is the point of a laptop, if you do so?)
disadvantages:
- it can only be used in one place and is more difficult to move around