There are hundreds of Linux distributions to choose from. One specific sub-category is a group of distros that advertise themselves as being lightweight or fast alternatives. They claim to offer better performance or to be better suited for older hardware. These statements are B.S. for the most part.
First thing to understand is that Linux is Linux. Fully featured, even the light ones, share the very same back-end architecture. You have the same Linux kernel with the same drivers, Xorg display server (soon Wayland) and so forth. Same software driving your hardware, getting the same performance out of it.
On top of the common Linux foundation you have the actual desktop and here we have some variance. Certain minimalist systems consume about 100-150 megabytes after booth, but they usually have a limited set of features. Distributions based on lighter, yet reasonably featured, desktops generally have a footprint around 300 megabytes. Here we are talking about Xfce, LXQt and so forth. Our example screenshot is from Manjaro LXQt and a footprint of 280 megs after boot.
Then to the big boys. The most visually and feature rich desktops Gnome, KDE and Ubuntu Unity all eat around 400-600 megabytes, depending on the configuration of the distribution. So the net difference to these lightweight alternatives is about 200-300 megs. This has very little difference on common hardware of recent years, where you most likely have 4 GB's of RAM or more. Even low end systems with only 2 GB RAM will leave you with quite a lot of headroom after first boot.
The savings of few hundred megabytes will have very little impact on your device's performance, with the only exception being that if you run completely out of RAM and the OS needs to perform swapping on the mass memory (HD or SSD).
The majority of your payload will not come from the operating system, or desktop but from the applications. Web browsers and sites are heavy these days and Chrome or Firefox will easily consume more RAM than your operating system and desktop altogether.
In the shot below we have Chrome running with 3 tabs, LibreOffice Writer with an empty document and GIMP without any image file opened. Combined they are already taking about 400 megs and the number will grow a lot once you get going with your work, whatever it is. So if you want to save some RAM, you need to choose lighter weight applications, but they are rarely full-featured.
The Phoronix performs periodical benchmarks for all parts of Linux and here is their recent report on Ubuntu 16.10:
For the most part, the performance between LXDE, GNOME, Openbox, and KDE Plasma tended to be about the same on Ubuntu 16.10 when using the stock Intel Skylake graphics support. The desktops that tended to be the slowest were actually Xfce and MATE for Linux graphics/gaming, at least in the out-of-the-box configuration as packaged in Ubuntu Yakkety.
For example, the KDE 5 is a very well made piece of software with excellent architecture and it performs very well even on older hardware, such as my Thinkpad X301 from late 2008 - hardware from almost ten years old! KDE developer Martin Gräßlin has written an excellent blog post regarding desktop compositing (the effects and such):
Without compositing we move the time-memory tradeoff towards using more CPU (create those windows) to prevent even more CPU usage. With compositing we don’t have to do anything of it. So disabling compositing is obviously not the silver bullet for being “lightweight”. It just means moving the time-memory tradeoff slider towards CPU.
So window compositors, like kwin or Compiz, offload some of the effort from the CPU to the GPU. It takes a bit of RAM, but reduces CPU cycles needed. So eliminating composition actually consumes more CPU cycles, loading it with work that the GPU should take care of, if you have one.
The lightweight and minimalistic distributions will give you simplicity and they save you a bit of RAM. If you prefer a simplistic desktop with less features then by all means, make use of them. Also, if you are seriously constrained with RAM, 2 gigs or less, then the savings will be worth while, as you are more likely to experience swapping.
They will not however make your system magically faster. For that you need to visit the local PC retailer.