Specular reflection is what occurs with a perfect mirror.
Diffuse reflection is what occurs with a theoretically perfectly matte surface (known as a Lambertian surface).
Neither of these exist in practice... they're the graphics programmer equivalent of how physicists like to work in a frictionless vacuum.
As mentioned by FLeBlanc, real life materials exist somewhere between these two extremes.
Generally, we model surfaces using a combination of the two; a diffuse-type reflection is used to model the refracted-and-diffused portion of the input light, and the specular-type reflection is used to model the portion of the light that reflects directly off a material without entering it.
This is nonsense to think of them in terms of pros and cons, because they're just different... What is this report that you're working on?
Diffuse - Models refracted scattering.
Specular - Models specular reflection.
Diffuse - Doesn't model specular reflection...
Specular - Doesn't model refracted scattering...
I guess if you're trying to model a theoretical material that's perfectly Lambertian (these don't exist, everything has a small specular reflection), and a material that's a perfect mirror (these don't exist either), and you have to pick just one of these models for some reason, then you could apply the above pro/con table to pick which one is most useful...
If you had to pick just one or the other, then you'd use diffuse for almost everything, and specular only for very shiny or translucent objects, like glass and metal... because usually the reflected portion of light is a much smaller percentage, so you'd get a better result modelling only the refracted light.
With pure metals, refracted light is never diffused/scattered (it's 100% absorbed), so you'd be better off only modelling the reflected light (which is therefore the only visible part).
With translucent materials, the refracted light isn't diffused/scattered, as it passes right through the object, so you'd again be better off just modelling the reflections.
But in most materials, the reflections will only account for <4% of the input light, so you'd be better off modelling the 96%+ of the light that is diffused.
There are many different diffuse and specular type models.
Lambertian is the most common diffuse model, and Phong or Blinn-Phong are probably the most common specular model.
There are many others though. For example, Oren–Nayar tries to improve upon Lambert's model to make it better approximate real world rough surfaces.
Likewise with specular, there's many better models than Blinn-Phong, such as Cook-Torrance.
As mentioned, any model of a real world material will use a combination of a diffuse and a specular model, blended together according to Fresnel's equations, which tells us how much light is refracted (and potentially diffused) vs how much is reflected. Again, in reality, both 'diffuse' and 'specular' occur, with the percentage of each ruled by Fresnel and the material's index of refraction.