Ohne Wasser werden die Pflanzen aussterben.
Ohne Wasser werden die Pflanzen vergehen.
Are aussterben and vergehen interchangeable?
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Your first sentence doesn't make sense.
to die out in the sense a whole species dies out, like for instance
Der Orang Utan ist vom Aussterben bedroht. Der tasmanische Tiger ist ausgestorben.
In your second sentence
vergehen could be used in that context, but I would rather use the word
absterben, which means
to die off
Ohne Wasser werden die Pflanzen absterben.
vergehen is usually used with events and time like
Die Zeit verging wie im Fluge.
Aussterben is simply to die out. You wouldn't use it for one single plant in your room in English and it works the same in German.
Vergehen means to pass (the intransitive version) and it can go in many directions one of which is to whither away. You can use it for pain or love and if you want you can also use it for plants (not for animals and humans). That would be a rather high brow use though. The common term for the basil plant on the window sill is eingehen.
So botton line... the two words have no overlap and they are never interchangeable.