JWST reveals its first direct image discovery of an exoplanet

How is it that we can spot a planet 110 light years away, but whether there’s another planet in the solar system past Pluto is a matter of legitimate scientific debate?

Because we are looking for much smaller planets.

Because exoplanets by definition are going to be found adjacent to stars, which limits the area you need to search. Planets are fairly common, so you don't need to look at that many stars before you find evidence of an exoplanet, provided you have a good-enough telescope.

A hypothetical planet beyond Pluto be in a huge part of the sky: Presumably the orbit of such a planet could be inclined about as much as Pluto's. The 17-degree inclination of Pluto's orbit means it could be in a 34-degree wide strip of the sky, which, if I'm doing my math right, is about 29% of the full sky. If we allow for up to a 30 degree inclination, then that's half the sky.

There's also the matter of object size and brightness. The proposed Planet Nine[1] was supposed to be a few hundred AU away, and around the mass of 4 or 5 Earths. The object discovered in this paper is around 100 M🜨, at around 52 AU from its star. Closer and larger. (Of course, there's a sweet spot for exoplanet discovery, where you want the planet to be close enough to be bright, but far enough away to be outside the glare of the star.)

1. https://en.wikipedia.org/wiki/Planet_Nine

The paradox is explained by different detection methods: exoplanets like this one glow in infrared and are directly visible against the black of space, while Planet Nine would be extremely dim, non-glowing, and lost in the cluttered background of our galaxy's disk.