The original meaning of よそ is "other place", so ~をよそに has a nuance of unconcern, and the main clause is always negative/bad things.
顧みる means "to take notice of/pay attention to/think of", and ~を/も顧みず(に) is the adverbial usage of a negative form. The meaning is close to ~をよそに, but positive/good things can follow this expression. That's the main difference.
e.g.
医者の忠告をよそに/顧みず、彼は酒を飲み続けた。
Disregarding the doctor's advice, he kept drinking.
彼は危険も顧みず、海に飛び込んで彼女を救った。(をよそに can't be used here)
Reckless of danger, he jumped into the sea and saved her.