In Buffy the Vampire Slayer and its ensuing spinoffs, Angel and Spike have always had a feud with each other. Regardless if they are fighting on the same side or happen to be enemies on opposite sides of the fence, these two can't stand each other. Beyond the fact that both men are two sides of the same coin, always destined to butt heads because they're too alike, there's never been a concrete reason for why they hate each other so much.
Episodes like Angel's "Destiny" rationalize why Spike hates Angel, explaining that it all began when the latter slept with the love of Spike's life, Drusilla, decades ago just for fun. As Angelus, he mentally tormented Spike as he would any other of his victims without reason, leaving Spike to blame Angel for him becoming a monster. The women in their life (Buffy included) could also explain Angel and Spike's constant bickering, but on paper, Angel doesn't have a reason to hate Spike as venomously as Spike hates him, yet he does.
However, Spike is able to make sense of why Angel returns the same level of hate for him. In Angel #8 by Bryan Edward Hill and Gleb Melnikov, Gunn asks Spike if Angel truly hates him, to which Spike responds that Angel, "hates his own reflection. When you're a vampire, you make people your mirrors." This definitively explains why the two hate each other so much.
Angel & Spike Hate Each Other Because They're Too Much Alike
If vampires are a reflection of each other, then this has to be especially true for Angel and Spike whose lives have paralleled each other in so many ways. As parallel reflections, Spike reminds Angel of the worst parts of himself that he used to be and, in some ways, still is. It's the exact same reason why Spike hates Angel.
Even years after Angel had already redeemed himself, Spike still sees Angel as Angelus whenever he looks at him. By proxy, Spike is a walking reminder of Angel's harsh past which he still feels guilty about. Spike, meanwhile, always jumps at the chance to remind Angel of his past wrongdoings and Angel hates him for it. Angel hates himself more than anything, but if he can only spend so much time beating himself up emotionally, then he can beat up someone emotionally and physically who reminds himself so closely of himself.
Angel and Spike have both pined for Buffy the Vampire Slayer, and they both are reformed vampires with valid reasons to hate each other.