Many younger people today have grown up without ever using a desktop computer at home. They have laptops/notebooks and do more communication and web access on smartphones. However, each type of platform as its own benefits and disadvantages, which are discussed in this post.
If you need portability from your computing device, a desktop computer isn't for you! Clearly, a smartphone offers the ultimate in portability, and notebooks do well too. The desktop doesn't stand a chance!
The price of portability, however, is that devices are easily lost, stolen or broken. Notebooks in particular are a top target for thieves, while phones are often lost. Portable devices are easily broken, by being dropped, sat on, stood on or someone tripping over the power lead. These problems don't apply to desktop PCs - even if your house is burgled, your desktop computer is unlikely to be taken these days.
Desktop computers tend to have more computing power than a notebook at a given price. The processors and memory will work faster, the hard drives will often spin faster and network links will often run at a higher speed. The hard drive is also likely to have more capacity. A smartphone will have far less computing clout, although phones aren't really designed for computational tasks so that might not matter too much.
Desktop computers typicaly have a longer working life than the smaller machines. A smartphone is usually worn out after 2-3 years and battery life will be hopeless. Notebooks also have shorter lives than desktops - battery life is again an issue after 2-3 years, and due to the components being crammed into a small space, they run hotter, reducing their lifespan.
An important and often overlooked factor is ergonomics. The small screen and keyboard of a phone are fine for a quick tweet, but impractical as a tool for sustained work. Notebooks are better, but the keyboards are typically not as good as desktop keyboards (the keys are often slightly smaller and don't have as much travel), and the relative positioning of the keyboard and screen is very bad for posture. Prolonged typing on a notebook over a period of time can lead to back problems and RSI (Repetitive Strain Injury) in wrists or shoulders. Desktop computer screens can be much larger than notebook screens, so you can fit more on screen or have everything displayed at a larger size, helping to avoid eye strain.
Finally, gamers will almost always choose a desktop computer. You can get far more games performance from a desktop at a given price, and can upgrade the desktop machine if necessary.
Anyone who wants to use their computer as a tool rather than a toy is likely to need a desktop computer rather than a notebook, unless portability is an important requirement.