afraid / anxious
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afraid
To have a feeling of fear, fright, or great worry. Often used in the impersonal expression 'jemandem ist/wird bang' (someone is/becomes afraid).
Mir wurde plötzlich ganz bang, als ich das unheimliche Geräusch im Keller hörte.
I suddenly became very afraid when I heard the eerie noise in the basement.
In the expression 'jemandem ist/wird bang' (someone is/becomes afraid), the person who feels fear is in the dative. The construction is impersonal; the subject is not the person themself.
Mir wurde auf einmal ganz bang ums Herz.
All at once, my heart became very anxious.
When you worry about someone or something, you use the fixed structure 'bang sein um' (to be anxious about) followed by the accusative.
Ich bin bang um seine Gesundheit.
I am anxious about his health.
The word 'bang' (afraid/anxious) belongs more to elevated or written language. It sounds somewhat more formal or poetic.
Dem Seemann wurde bang, als der Sturm aufzog.
The sailor became afraid as the storm came up.
A common and active use is the expression 'jemandem bange machen' (to frighten someone / to make someone afraid). It means to scare someone or make them afraid.
Der große Hund hat dem kleinen Kind bange gemacht.
The big dog frightened the small child.
brave / bold
'mutig' and 'bang' are opposites when it comes to fear. 'mutig' describes someone who acts despite fear or danger. 'bang' describes a feeling of fear or great worry. It is often used in expressions like 'mir ist bang' (I am afraid / uneasy) or 'ihm wird bang' (he becomes afraid / uneasy). So the difference is: 'mutig' emphasizes decisive action, while 'bang' emphasizes the anxious feeling.
Vor der Prüfung wurde Milan plötzlich bang.
Before the exam, Milan suddenly became afraid.
Die Ärztin war mutig und blieb auch in der schwierigen Situation ruhig.
The doctor was brave and stayed calm even in the difficult situation.