
das Armband
bracelet
A piece of jewelry worn around the wrist.
Sie trägt ein schönes silbernes Armband.
She is wearing a beautiful silver bracelet.
bracelet / wristband
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bracelet
A piece of jewelry worn around the wrist.
Sie trägt ein schönes silbernes Armband.
She is wearing a beautiful silver bracelet.

wristband
A band worn on the wrist for a specific purpose, for example as an entry ticket for an event or for identification in a hospital.
Zeig bitte dein Armband am Eingang vor, um auf das Festivalgelände zu kommen.
Please show your wristband at the entrance to get onto the festival grounds.
The word 'Armband' (bracelet / wristband) is a simple compound of 'der Arm' (arm) and 'das Band' (band). So it is literally a band that you wear on your arm.
Zum Geburtstag habe ich ein neues Armband bekommen.
I got a new bracelet for my birthday.
The most common verbs used with 'Armband' (bracelet / wristband) are 'tragen' (to wear), 'anlegen' (to put on), or 'ummachen' (to put on / fasten), and 'abnehmen' (to take off).
Er hat sein Armband vor dem Schwimmen abgenommen.
He took off his bracelet before swimming.
Context tells you whether a piece of jewelry or a functional band is meant. Words like 'golden' (gold), 'Perlen' (pearls), or 'Schmuck' (jewelry) point to meaning 1. Words like 'Eintritt' (entry), 'Krankenhaus' (hospital), or 'Festival' (festival) point to meaning 2.
Jeder Patient erhält bei der Aufnahme ein Armband mit seinem Namen.
Every patient receives a wristband with their name upon admission.
wristband / ribbon
Both words can refer to a band worn around the wrist to gain access to an event or for identification. Armband can be used for this, but without context it can also sound like a piece of jewelry. Bändchen usually makes the practical or identifying purpose clearer.

Am Eingang bekam jeder Besucher ein Armband als Eintrittskarte.
At the entrance, every visitor received a wristband as their admission ticket.
Am Eingang bekam jeder Besucher ein Bändchen als Eintrittskarte.
At the entrance, every visitor received a wristband as their admission ticket.