What is the difference between glossy and specular reflection? What is the difference between glossy and specular reflection?

What is their relations with BRDF?
The charts you show aren't showing two different phenomena - "glossy reflection" and "specular reflection" - they're showing two parameters of specular reflection. One is the specularity or specular colour and gives the amount or brightness of the specular reflection. The other is the glossiness or roughness and shows how sharp the specular reflection is.

Put another way, if you picture the curve of the specular reflection over changing angle, the specularity (the vertical axis on your charts) gives the height of the peak - the maximum brightness - and the glossiness gives the width of the peak - the size of the highlight. The highlight gets narrower with increasing glossiness, whereas roughness is the inverse of this.

BRDF is more general: it's a function that tells you how much light is reflected in a given direction (view direction) from a given direction (light direction). Usually (for most materials) we approximate a BRDF using a diffuse term and a specular term.