This topic is extensively discussed by astronomer, historian, and Marxist theorist Anton Pannekoek in his 1961 book A History of Astronomy on pp. 359-363 and his 1953 article "The Discovery of Neptune".
As he explains, the deviation in the calculated orbit and the observed orbit was 30" in 1835 and 70" by 1840.
LeVerrier and Adams both used these calculations and observations to predict an orbit of the hypothetical planet, but it is important to remember that these predictions contained many unknown variables that depended on the other variables, like mass of the hypothetical planet, its average distance from the sun and its eccentricity. To nevertheless compute an orbit, they used the at-the-time well-known Titius-Bode law that stated that the orbits of planets in AU followed the equation $$a = 0.4 + 0.3 \times 2^m$$
with m being -∞ for Mercury, 0 for Venus, 1 for Earth, 2 for Mars, 3 for Ceres, Pallas, Juno, and Vesta (which were considered planets), 4 for Mars, 5 for Jupiter, 6 for Saturn, and 7 for Uranus. It would therefore make sense that the new planet should have a semimajor axis of 38.8 AU (m=8).
Following LeVerrier's predictions, Galle of the Berlin Observatory indeed found a unmarked star in the region where LeVerrier had predicted the new planet would be. After tracking the star for months, however, astronomers quickly realized that the star was a planet but that it followed a different orbit than predicted. It turned to be smaller and nearer than the calculations predicted. Pannekoek illustrated this with a diagram:
This realization that the orbit was completely different than predicted led to a whole new discussion on whether LeVerrier and Adams had indeed discovered the planet, or that this was a new planet was a different object altogether.
Pannekoek goes into this discussion from a Marxist perspective and attempts to explain it by invoking the social circumstances of the various astronomers involved. France is in the middle of the 1848 revolutions and the bourgeoisie want to illustrate the power of the natural sciences against received wisdom, which explains why LeVerrier is so adamant at prompting Galle to find the new planet and why the French resist the argument that the calculated and the observed planets are different objects. Britain, meanwhile, already had a revolution in the seventeenth century and has the bourgeoisie already entrenched in government, which explains why Adams was only interested in theoretical astronomy and not so much the observational verification. The United States on the other hand had started as a bourgeois and democratic society where debate was important, so it made sense that American astronomers would challenge the idea that the calculated and observed planets are the same.
For discussions on Pannekoek's interpretations, see:
Robert W. Smith, "The Cambridge network in action: The discovery of Neptune", Isis: A Journal of the History of Science 80 (1989): 395-422
John G. Hubbell and Robert W. Smith, "Neptune in America: Negotiating a Discovery", Journal for the History of Astronomy 23 (1992): 261-291
Bart Karstens, "Anton Pannekoek as a Pioneer in the Sociology of Knowledge" in: Chaokang Tai, Bart van der Steen, and Jeroen van Dongen (eds.), Anton Pannekoek: Ways of Viewing Science and Society (Amsterdam University Press, 2019): 197-218