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.
Happy Reading.
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.
See this article from Phoronix for detail.
GNU will not adopt something as a project unless the developers agree to certain stipulations which bind all official GNU projects.
Currently the Linux kernel probably does not fit these restrictions, and there is nothing for Linus Torvalds, kernel.org, et al. to gain from placing themselves under the GNU umbrella, and a lot to lose -- the aforementioned binding agreement, and the public perception that the kernel is now a GNU project, which would have a mostly negative impact. GNU's parent organization, the Free Software Foundation (FSF), is a political organization and Torvalds has made various public criticisms of it and the somewhat controversial, iconoclastic lifetime leader/founder of GNU and the FSF, Richard M. Stallman.
Further, the Linux kernel does not require the GNU userspace any more than the GNU userspace requires the Linux kernel. This independence should be considered a good thing by the basic principles of software engineering, which favour modularity and looser coupling as opposed to the opposite (monolithic things with tight coupling).
Another point against this idea is that while HURD may not be of interest to as many people as Linux, the developers and users of HURD may object to having their project effectively dustbinned in a popularity contest. And good for them; "competition" of this sort is a positive thing, whereas bowing to monopolization is not -- you end up with massive entities that stifle creativity in part because they are prone to monolithic/meglomaniacal control. The Linux Foundation already is an independent organization, it might as well stay that way.
Best Answer
Assuming you're referring to binaries, no, you can't run GNU/Linux binaries on a GNU/Hurd system. Compatibility involves a number of factors, including the kernel; any binary which accesses resources managed by the kernel (including, notably, file access) will end up including kernel-specific information, such as how to invoke the kernel (syscalls etc.).
Calling a distribution “GNU/Linux” isn't a claim that it's compatible with the GNU operating system, in particular it's not a claim that it's compatible with a putative GNU operative system using the Hurd. It's simply a name which is meant to recognise the contributions of the GNU project. See Is it possible to run pure GNU? for more info on the GNU operating system, although these days the GNU project itself refers to Linux-based distributions as the GNU operating system (as long as they meet the Free Software Foundation's requirements).
You can build GNU/Hurd-based systems very similar to GNU/Linux-based systems; Debian GNU/Hurd is one example. You could also (theoretically) develop an emulation layer for Linux binaries on Hurd (similar to the iBCS2 compatibility layer for Linux in the past).