Can you pinpoint their subtleties for me please.
In general, Avert is used when taking action that will stop something happening. ie The pilot shut down the engine to avert a disaster

Avoid is used when not taking action which will stop something happening. ie He had to avoid eating nuts because of his allergy
See the synonyms here:

Avert isn't one of them.
For it, see:

However, you can avert your eyes in order to avoidseeing somethingEmotion: smile
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I think there are some cases when they mean the same: "to avoid/avert" an obstacle.

On the other hand, you can "avert" your eyes, but you "avoid" looking at something.
quite intresting.

I have never thought of them this way.

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 Marius Hancu's reply was promoted to an answer.
A last thought: you can also avoid somebody's eyes Emotion: smile