
die Müdigkeit
tiredness
The state in which you feel physically or mentally exhausted and have a need for sleep or rest.
Nach der langen Wanderung überkam ihn eine große Müdigkeit.
After the long hike, a great tiredness came over him.
tiredness / fatigue
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tiredness
The state in which you feel physically or mentally exhausted and have a need for sleep or rest.
Nach der langen Wanderung überkam ihn eine große Müdigkeit.
After the long hike, a great tiredness came over him.

weariness
A state of being fed up with or lacking enthusiasm for a particular thing or activity.
Nach Jahren im selben Job machte sich bei ihr eine gewisse Müdigkeit breit.
After years in the same job, a certain weariness began to spread in her.
The noun 'Müdigkeit' (tiredness) is derived from the adjective 'müde' (tired). The ending '-keit' is a typical way to form abstract, feminine nouns from adjectives.
Ihre Müdigkeit war nach dem langen Arbeitstag verständlich.
Her tiredness was understandable after the long workday.
'Müdigkeit' (tiredness) is an abstract noun and is normally used only in the singular. You speak of 'viel Müdigkeit' (a lot of tiredness) or 'großer Müdigkeit' (great tiredness), but not of several tirednesses.
Trotz großer Müdigkeit arbeitete sie weiter an ihrem Projekt.
Despite great tiredness, she continued working on her project.
In connection with physical exhaustion, 'Müdigkeit' (tiredness) is often used with verbs like 'verspüren' (to feel), 'fühlen' (to feel), or 'bekämpfen' (to fight). A common phrase is also 'von Müdigkeit übermannt werden' (to be overcome by tiredness).
Er versuchte, mit einem Kaffee gegen die Müdigkeit anzukämpfen.
He tried to fight the tiredness with a coffee.
When 'Müdigkeit' (weariness) describes being fed up with something, it often appears in the context of long-term activities or professions. Phrases like 'eine gewisse Müdigkeit' (a certain weariness) or compounds like 'Amtsmüdigkeit' (weariness of office) point to this figurative meaning.
Nach zehn Jahren im selben Job zeigte sich eine deutliche Müdigkeit im Team.
After ten years in the same job, a clear weariness appeared in the team.
exhaustion / depletion
Müdigkeit and Erschöpfung both describe a state of having little energy and wanting to rest. Müdigkeit is the more general state, while Erschöpfung refers to a much more severe lack of energy after considerable exertion. Use Müdigkeit when you simply need normal rest, and Erschöpfung after very intense exertion.

Nach dem langen Arbeitstag war die Müdigkeit groß, und sie ging früh ins Bett.
After the long workday, she was very tired and went to bed early.
Nach dem langen Arbeitstag war die Erschöpfung so stark, dass sie kaum noch sprechen konnte.
After the long workday, she was so exhausted that she could barely speak.