Laptops:
Pros (+) and Cons (-);
+ Portable
+ Peripherals
+/- Battery Powered
- Maintenance
- Customizability
- Power
- Price
Desktops:
+ Maintenance
+ Customizability
+ Power
+/- Peripherals
- Portable
- Socket Powered
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Let's look into this in more detail.
Laptops:
In regards of moving around and such Laptops are definitely superior to any Desktop you put it up against simply to to it's small size, lightweight build and battery life. In terms of Battery Life it also has the distinct advantage over Desktops in the fact that should you lose power in any way it can continue to be used until its power runs out. Be warned, consistent use of the laptop will cause the laptop battery to slowly degrade and eventually barely hold enough charge to last an hour. (Replacements are relatively easy to acquire)
Laptops are "All In One" considering that once purchased you do not need to then go out and buy a mouse, keyboard, speakers or monitors due to it all being in built. (Please Note: If you do wish to buy these the laptop is compatible with anything and everything Desktops are and as such you are not limited in any way,)
The issue with laptops and indeed with anything that you attempt to cram a massive amount of technology into a small amount of space the more compressed the technology the more power you lose and the greater the price. As such you tend to find to get the equivalent power of a desktop you can end up paying upwards of 1.5x the price of the desktop itself. In this same breath laptop maintenance is a difficult, time consuming and expensive procedure. Should you accidentally break a piece of hardware inside it it tends to be more expensive to repair than your generic desktop due to the difficulty of opening, accessing and replacing the component. Laptops also tend to be notoriously un-customizable due to the way that most are constructed. (I.e companies deliberately ensuring the components installed are the only ones that will fit.) As such to upgrade in the future you will have to purchase a whole new system rather than swap out parts.
Summary:
If you have a large budget, plan to travel a lot or do not intend to play high end games/stream/record then laptops may well be for you!
Handy, portable and excellent on the go I do advise using this for business work or if (For example) you wish to do some on the go editing/recording.
Desktops:
The major disadvantage of Desktops is the sheer bulk of the system. Transport is difficult and cumbersome, you cannot use it on the go (Without another special device) and they tend to be placed in a corner and never moved again. Adding to that fact you have the issue of purchasing peripherals (Monitor, Keyboard and Mouse at the very least) and possibly even backup peripherals to ensure that even if your mouse breaks you can keep using your system. Without any special devices the desktop also has absolutely no battery life meaning if you are hit with a power cut you lose all your work instantaneously. No warning, nothing.
On the other hand however, Desktops in general tend to be much cheaper and massively simpler to customize and maintain. With (Generally) large open cases that offer easy access, simple ways of upgrading specific components and more power per piece of hardware Desktops are classed as the more affordable, more powerful and much much more versatile brothers to the laptop. Want to play a game but don't get good frames? Upgrade your graphics card. Want to start streaming? Pop in a new CPU and roll away with it. This customisability leaves options open without having to purchase whole new systems (Unless you wish to make a massive upgrade.)
Desktops also tend to have much better heat management. This is especially key when doing anything that is "intensive" (Works the computer hard) as that can quickly cause heat to build up and can potentially damage components and cripple a system. Having more space between components and larger areas to fit extra fans goes a long way to keeping your system alive and fully functional for longer.
Summary:
If you wish to play high end games, get more power for less money, do any form of intensive work (Streaming / Recording) or simply wish to have something that can easily be upgraded and maintained for future use then I highly advise pushing into Desktop territory.
Stationary, powerful and versatile this system is best suited for someone who wishes to get more out of their system, does not travel much and wishes to play high end games without forking out more.
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Naturally some of the information in here (Summary in particular) is my own opinion but I hope this thread can assist you in choosing whether you wish to purchase a laptop or a desktop computer in the future. If I missed anything or you have any comments / questions feel free to leave a response and I'll get back to you ASAP.
Cheers,
~Icy