I was quite surprised to learn that the GNU project has an independent kernel of their own called Hurd. And there are mainstream distributions like Arch Hurd and Debian GNU/Hurd which uses it. Is there any significant advantage for Hurd over Linux?
At this point in time, considering there is no "stable" distribution of GNU/Hurd, the major advantages seem to lie with Linux.
A good place to start understanding the differences between a Mach microkernel, and a traditional monolithic unix kernel is the Wikipedia page on Mach (Kernel).
As an interesting note, Mac OS X, uses a Mach Kernel, called XNU. Though based on Mach 3.0, it's not a microkernel, like Hurd is. It makes sense, since Jobs brought the Mach kernel from NeXT to Apple when Apple bought NeXT.
Status Update 2018
In the first 6 months of 2018, the git repo for HURD received only 40 commits, so rumors of stability may be exaggerated. And the number of active code contributors is down to something like 5. So, GNU/Hurd is still at a major disadvantage to GNU/Linux. Check back in 2025 after another 7 years, for another update.
Hurd was the original 'anticipated' kernel BEFORE Linux existed. It has been under development, seemingly for years. During that time, Linus Torvalds along with volunteers worldwide, developed and implemented a kernel that is large, but worked. It was enhanced with programmers from many software firms including Redhat, HP and IBM. It is reliable and works. Mach adhere to the Minix philosophy of having a simple microkernel. I would suggest you take a look at the work of Minix if you would like to see the differences in philosophy. Though the microkernel may have some theoretical advantages, all of the literature I have read, seems to favor the kernel strategy taken by Linux.
According to the Wikipedia article on GNU Hurd:
In 2010, after twenty years under development, Stallman said that he was "not very optimistic about the GNU Hurd. It makes some progress, but to be really superior it would require solving a lot of deep problems", but added that "finishing it is not crucial" for the GNU system because a free kernel already existed (Linux), and completing Hurd would not address the main remaining problem for a free operating system: device support.
It seems that it would take a large amount of work to complete the project and resolve the issues that it is facing. On top of this, it is quite unclear what (if any) benefits the Hurd kernel would bring to most normal users over the current Linux kernel, which has had a large amount of resources poured into it over many years and works very well on a wide range of architectures.
Because of this, it seems that Hurd is little more than an academic exercise at this point. With no real driving need behind it, it seems unlikely that it will reach full usability any time soon, let alone catch up with or exceed the impressive capabilities of Linux. I don't mean to knock the Hurd developers, but I wouldn't hold your breath ...