When to use "essay" vs "assay" (as a verb)?

By the dictionary definition as a verb, I can see that essay and assay have the same meaning, i.e. "make an effort or attempt".

I'm wondering if they are totally interchangeable, or is there a difference in degree associated with each?

The modern meanings for essay and assay are almost unrelated. One essays a difficult task; one assays an ore or other material to objectively measure its value. A essayist is a writer; an assayer is a metallurgist.

Both have the same root, the French essai (trial). As verbs with the meaning of “attempt or try”, the New Oxford American Dictionary marks essay as “formal”, and assay as “archaic”. Apart from that, they are synonymous (and both antiquated).

Edit: regarding modern synonyms of this meaning, one would now say attempt or try. Other synonyms are endeavour, seek, undertake or have a go at.

The noun "assay" occurs in science, such as a "sequence assay", and scientists often verb such nouns (for example the verb "sequence", such as "we sequenced the human genome").

If you're talking about the verbed noun "assay", then it isn't interchangeable with "essay".