What is the difference between glossy and specular reflection?
What is their relations with BRDF?
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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.
Gloss and Specularity are features of the surface. In modern PBR terms we usually refer to the smoothness and metalness of a surface instead (unless you use a specular workflow, then metalness is still linked to specular).
Basically, dielectric surfaces reflect around 4% of light in a specular way, and the rest is diffuse. Metallic surfaces reflect no light diffused, and all specular. Specular light bounces directly back to the eye and shows clear reflections of the environment more. Diffuse bounces around a lot more and therefore gives a more matte result. Gloss is another term for smoothness of a surface. When a surface is rougher its specular light is bounced all over, and so the reflection appears more blurry.
TL;DR: Specular is how shiny it is and usually is higher with metals. Gloss is how clear the reflections are.
Keep in mind that there are some surfaces that don't fall into the metalness workflow, especially when they have a coat on top. There are also materials in between metal and dielectric (semiconductors / metalloids).
I've noticed a lot of confusion in the use of terminology in this field, sometimes also scientific papers or software packages misuse these words, I agree with @gallickgunner explanation.
To sum up and from my understanding:
materials have diffuse and specular reflection only.
in dielectric, you have both (and for physically based workflow they should never exceed 100%). If you have a certain amount of specular reflection you will have an inversely proportional amount of diffuse reflection and vice versa.
diffuse reflection is the light that scatters right underneath the surface and is reflected in all directions (diffuse reflection can be imagined as the base colour of the material or albedo), a material that has only the diffuse component looks flat and matte (like a very rough plaster), however, no materials in the world have only diffuse reflections.
the specular reflection is the light reflected on the surface of the material and can be sharp or blurry
the amount of blurriness of specular reflection is controlled by the glossiness parameter (or its inverse the roughness parameter)
for materials like metals the diffuse component is 0% and the specular is 100%, however, for coloured metals like gold or copper, the colour is given to the specular reflections and not to the diffuse reflections
a material that has 100% specular reflection with 100% glossiness, and 0% diffuse reflections it's a mirror
a material that has 100% specular reflection with 0% glossiness and 0% diffuse reflections looks like a very rough metal (like a heavily sandblasted silver) however, metals in real life always have some amount of glossiness it's never 0%
the "specularity" parameter is usually used to "weight" the % of specular reflection
the "specular colour" parameter is used to assign the colour to the specular reflections (metals and other strange hybrid materials can have coloured reflections; non-metallic/dielectric materials usually have white specular reflections)
the diffuse reflection might not always has a weighting factor as the specular reflections, usually it is controlled by the luminosity of its colour.
the "diffuse colour" parameter is the same as the specular colour but assigns a colour to the diffuse reflections rather than the specular ones.
the metalness does not describe a different type of reflection, but it basically makes sure that when increasing the specular reflections, the diffuse reflections are lowered proportionally. (so basically, when you increase specular reflections to 100%, even if there is a colour assigned to the diffuse colour channel you won't see it because it would be weighted down to 0%)
Do not mistake the diffuse reflection with the subsurface scattering (SSS), the SSS is an effect that scatters the light underneath the surface as the Diffuse, but it can go way farther underneath the surface compared to the diffuse, in theory, if you set the SSS distance to a very low number you could replicate the Diffuse reflection effect, however, it is much heavier to calculate, so when you don't need to go very far beneath the surface you just avoid calculating it and you rely only on the diffuse effect.
In the book Computer Graphics Principles and Practice, they use the term specular reflection when they want to imagine things resembling a mirror and glossy reflection when things like a polished door knob or an orange skin.
The charts shows you exactly that. When a material has more specular color, it should have less diffuse color due to the conservation of energy. That is, the sum of the light reflected specularly and light absorbed and emitted in random directions must be less than equal to 100% (the amount of light incident on the surface). Hence when you increase the specular color the material tends to go white or have a slight tint of the color like in metals.
Where as glossy surfaces can have more diffuse color but show a specular highlight like the surface of an orange skin.
So assuming the CGPP's point of view, we can say in pure specular reflection, the diffuse part is much less than the glossy part. Where as in glossy reflection the diffuse part is usually greater.