In the Arnold standard shader, you use specular reflections when you want glossy reflections. There’s direct specular (the “classical” specular hightlight) and indirect specular (glossy/blurry reflections), and both are part of the same BRDF behaviour. In this screenshot, direct specular (Direct Scale) is turned off, so you see only the indirect specular reflections. The Roughness makes those reflections glossy.
So what’s up with the separate Reflection/Refraction tab? Well, that tab gives you an alternate way to do sharp, mirror reflections (with no blurring or glossiness). However, with Specular, you can get pretty much the same thing by setting the Roughness to 0:
Finally, since I mentioned the direct specular, here’s the shaderball with just the direct spec:
Glad to be seeing some Arnold tutorials on here now. Congrats with the new position.