If you use over it can mean that the plane crossed the area where the clouds were, as if you say "the plane flew over the Sahara", while if you use above it means that the plane was higher than the clouds. Below (opposed to above) implies that something is anywhere in a lower position than another, anywhere. Under (opposed to over) denotes vertical line and covering (under a tree, under the bed) Beneath is an equivalent of both, below and under.
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