It's not a "moving image", it's a neural radiance field (NeRF), a way to represent a 3D scene using a neural network.

Style transfer worked on pixels, this approach works on the scene itself, before it's rasterized.

it seems like this is more stable. the style applies over a 3D world without noticeable artifacts or distortions

This was my impression as well. It seems to have some sort of object permanence when the camera is moved rather than just convoluting a single frame; seems significantly more difficult to me as a layman.

Yep. There has been some work to figure out frame-to-frame coherence on a sequence of images in 2D. But, I think this skips over that problem by working in 3D.