kaputt
broken
No longer working, broken, or damaged. Usually refers to objects or devices.
Mein Handy ist heruntergefallen und jetzt ist das Display kaputt.
My phone fell down, and now the display is broken.

broken / exhausted
Loading your saved state.
broken
No longer working, broken, or damaged. Usually refers to objects or devices.
Mein Handy ist heruntergefallen und jetzt ist das Display kaputt.
My phone fell down, and now the display is broken.

The adjective 'kaputt' (broken/exhausted) is very often used as part of the predicate with verbs like 'sein' (to be) or 'gehen' (to go). Attributive use before a noun, as in 'das kaputte Fahrrad' (the broken bicycle), is possible, but less common.
Das Glas ist kaputt.
The glass is broken.
Two very common verbs formed with 'kaputt' (broken) are 'kaputtgehen' (for something to break by itself) and 'kaputtmachen' (for someone to break something intentionally or unintentionally).
Die alte Waschmaschine ist einfach kaputtgegangen.
The old washing machine simply broke.
The context reveals the exact meaning: if 'kaputt' (broken) refers to an object, it is broken. If it refers to a person, they are exhausted. If it refers to something abstract like a relationship, it has failed.
Der Fernseher ist kaputt.
The television is broken.
When 'kaputt' (exhausted) describes a person who is very tired or worn out, it is used colloquially. In more formal contexts, you would rather say 'erschöpft' (exhausted) or 'ermüdet' (fatigued).
Nach der Arbeit bin ich oft völlig kaputt.
After work, I'm often completely exhausted.
except / besides
'außer' and 'kaputt' are similar only in a specific context: in the expression 'außer Betrieb' (out of service). When a vending machine or elevator is 'außer Betrieb', it is currently not working or not being used. 'kaputt' says more generally that something is damaged or no longer works. 'außer Betrieb' often sounds more formal and is frequently seen on signs for devices or facilities. 'kaputt' is more natural in everyday life when simply talking about an object.

Mein Handy ist kaputt und lässt sich nicht mehr einschalten.
My phone is broken and won't turn on anymore.
Der Fahrstuhl ist heute außer Betrieb, deshalb nehmen wir die Treppe.
The elevator is out of service today, so we're taking the stairs.