I have been told by some friends that Raspberry Pi computer is the "next best thing".
This is an ambiguous, unqualified, and subjective statement -- if my favourite thing is racing cars it is very unlikely I would consider the Raspberry Pi "the next best thing", etc.
So to be meaningful and usefully evaluated, we need to consider some further context.
If I already have a desktop running Linux, can't I already do all the Pi projects with my desktop? [...] but functionality-wise, isn't my Linux desktop just as good as the Pi?
Yes and no. Or maybe no and yes would be the correct order, in the sense that:
No, you cannot do "all the Pi projects with your desktop". However, all or most of the things that you can't do with your desktop may in fact be things that you are not interested in doing. (There is a bit of a qualification to this which might make the answer "yes", which I'll get to).
Yes, your desktop is "just as good as the Pi" and in fact is certainly way, way, way better than any current model of Pi unless you have an antique desktop. However, here I'm considering the standards people evaluate normal computers by, such as processor speed, memory, bus types, etc.
The Pi is not a replacement for a desktop and anyone who tells you this is delusional and/or a particular sort of iconoclast. I'm not using the latter in a pejorative sense, BTW; if you are a similar sort of iconoclast you may just love it. But most consumers will be disappointed in this context.
So why would I want one?
There's any number of reasons for that, but most of the major ones have to do with the GPIO breakout. For example, this includes access to an I2C bus. Your desktop probably has one for purposes listed in that wikipedia article (e.g., working with the real-time clock on the motherboard) but it is not something you can find by looking inside and it certainly won't be anything you can make much use of.
This is not true on the pi, where the two serial lines which are the I/O for the bus are directly accessible and can be attached to a wide variety of sensors and devices in a form that is about as small and inexpensive as such devices can get. I mentioned earlier that "there is a bit of a qualification to this" regarding your desktop, because you can, in fact, buy a USB to I2C adapter and do the same kind of things -- except these adapters tend, as far as I'm aware, to cost at least as much if not substantially more than a Raspberry Pi.
There's also power consumption. A pi will use < 5 watts sans peripherals. So if you have a purpose for a computer you want to leave on somewhere 24/7, that is a major advantage over your desktop, which will use at least an order (or two) of magnitude more electricity, take up a lot of space, and probably be as silly (or eccentric) in comparison as using a pi for a desktop. Any model of pi will match the power and potential of most home routers, NAS systems, etc., with the important caveat that its network throughput is more limited. Put another way, they make great general purpose home servers.
They are also used as digital media/entertainment centers (which is sort of a combination of home server and lightweight GUI computer), primarily via Kodi. There are various arrangements along this line depending on how tech savvy you are; my personal use case here revolves around an FM transmitter attached to the aforementioned GPIO pins. Beware that the analog output sound system is not very high quality, but there are ways around that (such as streaming to another device and using an external adapter).