The club's Latin motto Nil Satis, Nisi Optimum, which means "nothing but the best is good enough", and two wreaths have been removed from the new design.
The petition said fans would not buy merchandise bearing the "awful" crest and this would cost the club money.
Everton said it was "sorry" it had not properly consulted with fans.
However, the Premier League club said the new badge would be used during the 2013-14 season.
The club said: "We regret we didn't ask every Evertonian about something that matters so much to every one of you."
But it said it remained "firm in the belief our crest should be modernised".
The motto was added to the club's crest in 1938, though it did not appear on kit until 1978.
It was removed from the kit badges in 1982 but returned to the crest after a redesign in 1991.
Comments on the petition described the new badge 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 on Sunday.
The club said the fan reaction had prompted "several meetings and countless discussions".
Everton Fans' Forum, an "excellent and effective group of representative fans", was consulted on the badge redesign, but faced "too big a burden speaking for the entire fan base on something so significant", the club said.
Robert Elstone, Everton's chief executive, said: "Our chairman had demanded widespread consultation and we stopped short of that.
"We talked to our Fans' Forum, our commercial partners and our experienced staff around the club. That was not enough."
He said the club would seek fans' views over its badge for the 2014-15 season, consulting with a panel "from all sections of the fan base".
The club said Evertonians would "make the final decision".
Details on how the panel will be selected are expected this year.
Everton have apologised to fans after more than 22,000 people signed a petition criticising the football club's new badge.