need / hardship
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hardship / distress
A very difficult situation in which someone lacks the basic necessities of life, such as money or food. It can also describe a general situation of great danger or distress.
Durch die Überschwemmung gerieten viele Familien in große Not.
Because of the flood, many families fell into severe hardship.
The word 'Not' (need/hardship) is a very productive component of compound nouns. In that case, it usually means 'for emergencies' or 'urgent'.
Im Brandfall benutzen Sie bitte den Notausgang.
In case of fire, please use the emergency exit.
'Die Not' is a German noun and means 'distress' or 'misery'. It has nothing to do with the English negation word 'not', even though the spelling is identical.
Viele Menschen litten nach dem Krieg große Not.
Many people suffered great hardship after the war.
In connection with poverty or danger, 'Not' (need/hardship) is often combined with certain verbs. Typical phrases are 'in Not geraten' (to fall into hardship), 'Not leiden' (to suffer hardship), or 'jemandem in der Not helfen' (to help someone in need).
Durch die Krankheit ist die Familie in finanzielle Not geraten.
Because of the illness, the family fell into financial hardship.
The fixed expression 'zur Not' (if necessary/as a last resort) can stand at the beginning of the sentence (position 1) or in the middle of the sentence. If it stands at the beginning, the verb follows directly.
Zur Not können wir auch zu Fuß gehen.
If necessary, we can also go on foot.
The expression 'seine Not mit etwas haben' means 'to have difficulties with something'. The possessive pronoun ('seine', 'ihre', 'meine') changes to match the person who is having the difficulties.
Ich habe meine liebe Not mit der neuen Software.
I'm really struggling with the new software.
effort / trouble
'die Not' and 'die Mühe' can both express that something is difficult. But the difference is important: 'Mühe' mainly means effort or difficulty with a task. 'Not' is stronger and normally means a serious situation in which someone needs help or something important is missing. They are especially easy to confuse in phrases like 'seine Not haben mit' (to struggle with) and 'Mühe haben' (to have difficulty). There, both mean that something is difficult. In normal situations, however, you are more likely to say 'Mühe' when talking about learning, work, or understanding. 'Not' fits more with great distress or in fixed expressions.
Tom hat seine Not mit dem neuen Computerprogramm.
Tom is struggling with the new computer program.
Meine Nachbarin hat Mühe, die schweren Einkaufstaschen zu tragen.
My neighbor has difficulty carrying the heavy shopping bags.