TL;DR: There was apparent 11% increase of Neptune brightness during 1980 and 2000. This could be due to multiple reasons. Recent observation suggested the reason to be change in the amount and brightness of the banded cloud mostly in the planet's southern hemisphere due to seasonal variations and thus affect overall brightness of the planet.
Neptune's visual magnitude stayed relatively constant before 1980 and after 2000, but during this period, there was an unexpected rise in the brightness, about 11%. Astronomers have proposed various ideas to explain this peculiar variation but none are convincing. But the more promising explanation would likely to be seasonal change due to tilt and cloud formation variation.
A team from Wisconsin made three sets of observations of the full rotation of the planet in 1996, 1998, and 2002. The images showed progressively brighter bands of clouds encircling the planet's southern hemisphere. The findings were consistent with observations made by G.W. Lockwood at the Lowell Observatory, which show that Neptune has been gradually getting brighter since 1980. They believe that cloud bands are getting brighter because the Sun is warming the atmosphere in the south apparently changing the brighness of Neptune.
Neptune's near-infrared brightness is much more sensitive to high altitude clouds than its visible brightness. The recent trend of increasing cloud activity on Neptune has been qualitatively confirmed at near-infrared wavelengths with Keck Telescope observations from July 2000 to June 2001 by H. Hammel and co-workers which could provide an explanation to this brightness change.
Tilt could also be a reason. Hubble images reveal a real increase in Neptune's cloud cover consistent with seasonal change is the apparent absence of change in the planet's low latitudes near its equator. This do has a reason. Since Neptune is inclined at a 29-degree angle, the amount of solar energy received determines the season and this seasonal variation might affect the brightness. When the Sun deposits heat energy into the atmosphere, it forces a response like heating the atmosphere that lead to constant moving and condensing of cloud cover.
Another reason could be internal heat source of Neptune which may contribute to the planet's apparent seasonal variations and ultimately affect the planetary brightness.
Current visual magnitude of Neptune is 7.88 but it could reach 8.0 in the near future.
For more information, refer to this paper, "The Secular and Rotational Brightness Variations of Neptune"