Hmmm, let's see
The Beast has a slight edge in power, for having broken the moon into a radial third in the early days of its awakening. He only got stronger from there, and he chased Kessler for a few years.
Hax: Ragyō could oneshot with Mental Refitting, but the thing is, I'm not sure if The Beast even has a physical brain. As we saw from the battle with Cole, the inside of his head is filled with a lava-esqu energy. There may not be a brain for Ragyō to attack.
The Beast on the other hand, has numerous methods of keeping the threads away: can release an omnidirectional blast, use his matter absorbing black holes to eat them, or maybe use his gravity manipulation to help out.
His Ray Field Blasts are extremely deadly, since they basically kill anything without a Conduit gene. Ragyo will likely survive the radiation, since she's a life fiber hybrid, but the blast itself may kill her. It starts off by first draining her of her nuero electricity, converts it into Ray Field energy, and then releases it in a destructive blast.
Beast also has power absorption going for him. Though it requires physical contact, once he starts, it's difficult to fight off. He drains you of your powers, breaking your abilities and taking away your techniques. He did this to Cole in the beginning of inFAMOUS 2.
And if Ragyo touches one of his pseudo black holes, it's basically game over. It instantly breaks down and absorbs whatever matter touches it.
Tl;dr Beast via regen, more hax options and likely resistance to her specific form of mind manipulation
