I think it's possible for these two organizations to exist in the Bond Universe.
Quantum struck me as a
business. These were capitalists involved in numerous things (and not always evil things) with the sole intention of making money. Sure, human life wasn't that important to them, but their main goal always seemed to be "how can we make more money?"
SPECTRE always seemed to me to be about money
and power. Their goals were to create global chaos and use blackmail as their methods of seizing money. SPECTRE wanted to extort money from governments. Quantum used their power to destabilize regions because it was 'good for business.' That's all this was to them: business. Think Halliburton....but with no morals.
It's possible that Mr. White may have left Quantum for SPECTRE.....or he may even be in both. Still using his Quantum membership to make money while using SPECTRE because he can make even more money.