The difference between atheism spreading hate and religion spreading hate is that there's really no way to get hate just out of the definition of atheism or what being one entails (so atheism by itself doesn't spread hate, rather personal beliefs/morals/standards etc coupled with atheism can, or rather, coupled with pretty much anything), whereas it would certainly not be impossible for a follower of a religion to justify their hate through that religion's holy/sacred book, either by it's promotion of violence to other groups, intolerance, or a multitude of different factors (of course this would depend on what religion).
If you're asking if atheism itself can spread hate, the answer is "no" because atheism is a lack of belief without specific hateful beliefs stitched into its theology. Lack of belief is morally neutral- whereas a religion that specifically teaches hateful commandments is morally hateful. For example, look at the Old Testament in the Bible and you'll notice it has verses where it specifically commands hateful things. The members of this religion may choose not to follow hateful commands and it can show that they, themselves, don't spread hate, but their religion itself still does! Atheism can't spread hate, but atheists definitely can and some often do.
Since atheism is the lack of belief in religion then simply being an atheist cannot make you spread hate. Atheists don't congregate and follow set ideals or rules like religious people do. Being an atheist doesn't mean you lack morals or compassion. It just means you don't need a higher power to scare you into being moral or compassionate. Since atheism has been around as long as religion and there have been no instances of mass genocide by atheism and plenty of instances of mass genocide because of religion I think that answers your question pretty accurately.