Learn Russian: Искусство иллюзии — отнюдь не обман, а просто ловкость. - The art of illusion is by no means deception, but simply dexterity.

While all three mean 'by no means' or 'far from', отнюдь не carries a slightly more formal, literary, or intellectual tone. Because the sentence discusses an abstract concept like искусство (art), this formal tone fits perfectly. Вовсе не would also work but sounds a bit more everyday and conversational.

I thought we drop the dash between two nominative nouns if there is a не. Why is there a dash before отнюдь не?

You have a good memory! Usually, the dash replacing the present tense 'is' is dropped if the second part of the sentence starts with the negative particle не. However, writers often keep the dash if they want to emphasize a strong logical pause or contrast. The formula не X, а Y makes this contrast very strong, which justifies keeping the dash here.

Could we use но instead of а for 'but' in this sentence?

No, а is strictly required here. In Russian, when you completely negate one thing and replace it with another (not X, but instead Y), you must use а. You would use но if you were adding a contrasting but non-exclusive fact, rather than a direct replacement.

Why is it Искусство иллюзии (art of illusion) instead of using an adjective like иллюзорное искусство?