Some of what Seutonius repeats is scandalous and of dubious veracity, notably his clear imputation that Nero was culpable in the great fire of Rome. At the time the rumor was that he had done so in order to clear space for his palace, but his subsequent behavior in providing housing and food for the survivors leads us to take this with a grain of salt. His persecution of Christians as scapegoats in that fire was opportunistic and cowardly, but not, evidently, done out of any particular animus for that new religion. Nor did it get him off the hook, as a glance at Seutonius shows.
For those interested in commentary on Seutonius I'd recommend Michael Grant's The Twelve Caesars.
It was Nero who called Christians "enemies of mankind." That sounds like a pretty particularized animus to me.
Nero's anti-Christian animus is repeated often here, usually by gay activist athiests. That isn't surprising given that they and Nero speak from a common spirit.