Question: *Plesionyms* are synonymous words which have slight differences in meaning.

What are the examples of it? I found:

* Fog v Mist
* Fearless v Brave

When and why are they are used?

What are the aspects which differentiate plesionymic synonyms from cognitive synonyms?

Answer: 
When words have a similar but slightly different meaning, there will be contexts where one is more appropriate than another. In other contexts, the difference may not be relevant so either is acceptable.

In your examples: A brave person and a fearless person may both perform the same feats that others may be too fearful to perform. However, a fearless person would not be afraid to perform the feat whereas a brave person may perform the feat despite being afraid. From an outside observer's perspective, if the person did not appear to be afraid, then the observer might use brave or fearless to describe the person. If, however, it was clear that the person was afraid but did it anyway, then only brave would make sense.
It depends on the context.

In a nearly polar land with frequent precipitation there will be many words used for **snow** and they aren't all synonyms: they describe different types of snow which are relevant to the environment.

But in an equatorial land where it never snows, the words **snow** and **sleet** and **hail** might be considered to be synonyms.