Dear speakers of Russian, please explain me the lexical difference between the words “необычный” and “необычайный” as to me they seem pretty much the same.

For me no difference, just synonyms.

Very close synonyms, of course. However, НЕОБЫЧНЫЙ is just “unusual” whereas НЕОБЫЧАЙНЫЙ has a bit more mysterious coloring.

“необычайный” is always positive while “необычный” may also have a negative connotation:

  1. С нами произошла необычная история. - An unusual story happened to us. (there might be something bad that happened to us)
  2. С нами произошла необычайная история. (An incredible story happened to us.)

also:

  1. У него необычный дар. - He has an unusual gift (this gift might be dangerous or harmful)
  2. У него необычайный дар. - He has an exceptional gift.