The main changes comparing old with new - which at first glance look pretty similar is that he club's motto - Nil Satis, Nisi Optimum - and a pair of wreaths have been removed from the badge.
An Everton spokesman said it was "more modern, cleaner and dynamic".
However, the petition states the "awful" crest would lead to losses "due to no-one wanting to buy any merchandise with that printed on it".
The Latin motto, which translates as "nothing but the best is good enough", was added to the club's crest in 1938, though it did not appear on a kit until 1978.
It was removed from the kit badges from 1982, but returned to the crest following a redesign in 1991.
Commercial director Dave Biggar said the club had met with "various fan entities" including "supporters' clubs, fans' forums [and] representatives of shareholders" during the design process.
He said the idea had been to create a "logo device which is easily recognisable, easily replicated and easily communicated".
Fans' forum chairman Tony Bott said he had been won over by the new design, although comments on the petition described it as "amateurish", "poorly designed", "ridiculous", "clownish" and "not fitting a club of Everton's rich history and stature".
Merchandise using the new design, which is the 10th version in the club's history, went on sale for the first time on Sunday.