die Härte
severity
Great strictness or relentlessness in acting or judging.
Der Richter entschied mit aller Härte des Gesetzes.
The judge ruled with the full severity of the law.

hardness / severity
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severity
Great strictness or relentlessness in acting or judging.
Der Richter entschied mit aller Härte des Gesetzes.
The judge ruled with the full severity of the law.

'Die Härte' (hardness/severity) is the noun corresponding to the adjective 'hart' (hard). All meanings of the noun derive from the basic idea of 'being hard', whether with materials, in character, or in a situation.
Der Stein ist sehr hart.
The stone is very hard.
In the meaning of 'Strenge' (strictness/severity), 'Härte' (severity) is often used in formal or legal contexts. A fixed expression is 'mit aller Härte' (with full severity).
Die Polizei kündigte an, mit aller Härte gegen die Kriminalität vorzugehen.
The police announced that they would take action against crime with full severity.
When it is about the physical property, you will often find the word 'Härte' (hardness) together with materials such as 'Stahl' (steel), 'Glas' (glass), or 'Diamant' (diamond), or in technical descriptions.
Die Härte von Stahl kann durch verschiedene Verfahren erhöht werden.
The hardness of steel can be increased through various processes.
When 'Härte' (toughness) describes a character trait, the verb 'zeigen' (to show) is often used. The phrase 'Härte zeigen' means that someone is emotionally strong and resilient.
In dieser schwierigen Verhandlung musste sie Härte zeigen.
In this difficult negotiation, she had to show toughness.
The colloquial meaning is often used as an exclamation: 'Das ist die Härte!' This expresses that you find something unbelievable or outrageous.
Er hat mein Fahrrad genommen, ohne zu fragen. Das ist die Härte!
He took my bike without asking. That's outrageous!
difficulty
'die Schwierigkeit' and 'die Härte' can both express that a situation is difficult or unpleasant. 'Schwierigkeit' is neutral and means a problem or obstacle. 'Härte' in this meaning is more colloquial and stronger. It emphasizes that something is very exhausting, stressful, or hard to endure. Learners may confuse the words because both can fit difficult situations. If you are speaking matter-of-factly about a problem, use 'Schwierigkeit'. If you want to emphasize how hard or unpleasant something was, 'Härte' is more suitable.

Zehn Stunden Arbeit ohne richtige Pause waren für das ganze Team eine echte Härte.
Ten hours of work without a proper break was a real ordeal for the whole team.
Beim Lernen ist die Aussprache für mich im Moment die größte Schwierigkeit.
When learning, pronunciation is currently the biggest difficulty for me.