The 14th amendment is a restriction on government. Civil rights acts on private entities involved in intrastate trade.

CRA also expressly allows racism unlike the 14th. For instance the civil rights act allows a business to preferentially treat an Indian in a near reservation (but off res) business while outlawing discriminating say in favor of blacks near a historically black neighborhood.

But the 14th amendment didn't actually work in practice. Nearly 100 years later and many states had explicitly racist laws and policies - in some cases supported by the Supreme Court - and those that the 14th amendment were supposed to protect were disenfranchised.

If the 14th had actually worked then the Civil Rights Act wouldn't have actually been needed, but there is a lot of hard evidence that it, or something like it, was needed.