tl;dr: Always use GGX, set your roughness map to Non-color Data and square it with a math node before plugging it into the Glossy BSDF.
Different microfacet distributions will have differently shaped specular highlights. Beckmann and Ashikhmin-Shirley are both similar to a Blinn-Phong specular highlight that you might find in a game engine or Blender Internal, although they're normalized so that tighter highlights are brighter and duller highlights are less bright.
If you are just rendering for yourself I'd recommend sticking with GGX unless you've got a really, really good reason not to. The longer tail on the highlight makes the microfacet distribution more believable. Generally people remap the roughness map to make it more perceptually linear, so that if something is 30% glossy (for example) a specular highlight will have a radius of 30% of the radius of the hemisphere. Renderman, Unreal Engine 4, and Substance Designer and Painter all square the roughness for GGX, while Cryengine uses the formula
roughness=(1-smoothness*0.7)^6 (Schulz 2014) because Cryengine loves to be weird and do things differently. Finally, Marmoset Toolbag 2 and Unity both use a normalized Blinn-Phong BRDF which is most accurately reproduced in Cycles by plugging the roughness through an RGB curves node and plugging that into a Glossy BSDF set to Ashikhmin-Shirley. Here's a chart with all of the different microfacet models and remappings. Note that all of these assume you read your roughness map either from an alpha channel or from a texture that's marked as Non-Color Data! Otherwise, the remapping is going to be different from what you probably intended.
From left to right, the roughness or glossiness parameter is increasing from 0 to 1 in increments of 0.1. From top to bottom, they are vanilla Beckmann, vanilla Ashikhmin-Shirley, vanilla GGX, GGX with roughness squared (like UE4 and Renderman), GGX with the roughness squared and inverted for easier comparison to Cryengine's remapping, the Cryengine GGX remapping itself, and the Unity/Toolbag remapping of Ashikhmin-Shirley. Here they all are again with more reflection bounces and an environment:
Beckmann acts particularly oddly at higher roughnesses (>.6) so if you're going to use it I would stay away from making your materials rougher than that. Ashikhmin-Shirley looks better, but GGX looks the best in my opinion. As far as remappings go I think Crytek's is the most linear-looking but I prefer to use the UE4 mapping as Substance Painter and 3D-Coat both support it and it blends well enough between material layers with different roughnesses.
The RGB curves node that remaps Unity/Toolbag 2 smoothness to Ashikhmin roughness looks like this if you're interested. I just eyeballed it but it's pretty close:
The points are as follows: (0,0), (.13182, .05625), (.45455, .18750), (.81364, .36785), (1, .61250). This inverts the roughness map (just like Toolbag and Unity) and compresses it to a region where most of the plausible surfaces are, as well as making the highlight's size roughly linear as smoothness varies, making texture authoring more predictable.
Here's the node setup for the Crytek roughness remapping that converts smoothness to GGX roughness.
The UE4/Renderman/everything else roughness remapping is easy to set up:
Note that in all of these cases a 100% rough glossy BSDF still doesn't look the same as a Diffuse BSDF. Because the BRDFs that Cycles implement always reflect the light depending on the half angle, rough glossy BSDFs will look really rough but will still look like rough metal instead of a rough dielectric.