What is the difference between Arch Linux and Gentoo Linux? Their ideologies seem quite similar to me.
Yes, the distros are of similar, with both being set to satisfy more experienced users, and both aim to be fast and highly customizable. Th most technical similarity is that both are based upon the Linux Kernel.
While most functions may seem similar, the two are different in many ways.
Apparently, Gentoo documentation is said to be very intimidating to new users, while Arch documentation is very much up to the KISS (Keep it simple, stupid) motto.
Package managers are also different. Arch Linux uses the Pacman (or in some spins, such as antergos, Pacman XG) which uses the good precompiled package system while Gentoo uses the Portage manager which makes packages from source code.
With the difference in package managers one distribution may have fewer packages readied than the other. I would say that Arch would have a larger selection of packages compared to Gentoo, but the selection of individual packages may be different as well.
However, most packages are available in source code. So you can fairly easily build them to suit whatever package manager you may be using.
(If you may be interested, Gentoo's portage manager has many good features not available in the freshly-installed pacman)
Popularity is a difference. While you may be interested in being original, the adoption of your OS can make a big difference in your Linux experience. Primarily in how many files you can access out-of-the-disk and how many tutorials you have to look at in times of need.
According to distrowatch, Arch Linux is the 8th in overall popularity, while Gentoo is at 47th.
While popularity may help, this may not help you to easily choose a distro. I haven't personally tried Gentoo, it could just be an amazingly functional and simple OS, while Arch had risen up much further with its head-start.
Gentoo has a good variety of officially supported desktops from AfterStep and BlackBox, to Gnome and Xfce. Arch Linux officially supports mostly the major Desktops. (Probably because it is popularly adopted as a command-line system).
I could list many more differences, but aside from the above (and maybe other) differences, the distributions are quite similar.
If you would like a good resource to make comparisons with, I recommend distrowatch.com, if you haven't looked at it already.
Probably the biggest difference is that gentoo provides source packages while arch provides pre compiled binaries. Arch also only supports x86 machines although it has been ported to other architectures with some success. Look here for a list.
In the installation of Gentoo, you'll be much more encouraged to compile your own kernel, an experience that any power user of Linux should go through. :)
Arch by default uses systemd for its initialisation. systemd is growing much more popular, and most distributions are moving over to it in place of the old System-V style init system. Gentoo uses this older init system by default, but is aided by OpenRC. Gentoo however does have systemd available in Portage.
There's one similarity that I'd like to mention though: you can learn a lot about Linux just from installing either distribution!