I have translated loneliness into German. There are three translations for that. What is the difference between these three words?
Einsamkeit is the most general of all the nouns in the question. It has the most general meaning of the state of loneliness, i.e. nobody else being around. What the reasons are is irrelevant, Einsamkeit can be used.
Beduinen ziehen tagelang durch die Einsamkeit der Wüste.
Vereinsamung refers more to the process of isolation and is generally used in social (science) contexts. Often the sentences it is used in imply pity for the people being isolated or isolating themselves (but that is not necessary).
Erst kommt der Alkohol, dann die soziale Vereinsamung.
Verlassenheit is a word I have rarely encountered to date, but dwds.de provides sufficient proof of its existence. It is derived from the adjective verlassen which means left alone (or the correspondig verb to leave). Contrary to Vereinsamung, which describes a person’s state, Verlassenheit would more typically be used to describe their feelings.
Nachdem die Anderen gegangen waren, ergriff ihn ein Gefühl der Verlassenheit.
Einsamkeit is a state not referring to anything.
Verlassenheit means you where not lonely before, so others left you alone. Both are like a feeling.
Vereinsamung describes a social isolation (maybe like the loneliness of more persons). It is more a science word I think.
The common translation would be
I have read an article about this topic. In this article, it differentiates Einsamkeit from Alleinsein.
The article basically says that these words seem very similar but actually mean very different at their cores.
Alleinsein is what you understand from loneliness. Even if you want some people around you but because of some reason, this does not happen and you are alone.
In contrast to Alleinsein, Einsamkeit means that you have people (your family, your friends, your colleagues etc.) around you but you still feel yourself alone. You think and/or feel that they do not understand you well.