Yes, the difference being that the existence of Neptune and Pluto could be visually confirmed through telescope more easily. They’re also not visible to the naked eye, but they can be found in telescopic images by comparing to a map of the sky at other times. Later on they were visited by probes.
I don’t know what counts as a discovery of a planet. Personally, I’d like to see any kind of real data from the planet before claiming it discovered.
The calculations can be 100% correct, but they’re not verified until there’s some kind of external proof.
The planet X (or 9 or whatever) has yet to be seen in any kind of way.
So far the calculations have been confirmed by other calculations showing the same thing, and they’re most certainly correct, but the “solid” proof is still missing.
(It doesn’t have to be solid or visual, I don’t know the English word for it; it just needs to originate from the actual existence of the planet and not only from the effects of its theoretical existence.)