The Great Red Spot is the iconic feature of Jupiter, just as rings are for Saturn. With a diameter simillar to that of Earth, it's huge! Detailed observations by telescopes and the Galileo spacecraft show that it is an enormous and powerful storm, with wind speeds of up to 600 km per hour. And most amazing of all, it has been raging for centuries – it has been continuously observed since the late 1800's, and the spot observed on Jupiter by Hooke and Cassini in the mid 1600s may well have been the same storm, which would make it over 350 years old at least!
Clearly something is changing in the spot, but scientists are unsure of the reasons, as there is no consensus on the processes that have formed it in the first place.
Neptune's atmosphere is still very much a mystery and is likely to remain so for a long time, because no space missions in that direction are planned for the foreseeable future, the busy Hubble telescope can only be focussed on Neptune infrequently and with the planet being so much more distant than Jupiter, ground-based telescopes cannot pick out enough detail.
But it's a different story for Jupiter's Great Red Spot - NASAs Juno spacecraft is due to reach Jupiter in July 2016, and one of its goals is to investigate variations in the atmosphere there, so perhaps we will not have to wait too long for more light to be shed on this mystery.
Shrinking Great Red Spot http://phys.org/news/2015-10-great-red-orange-pimpleis-jupiter.html
Spots on Neptune http://www.nytimes.com/2014/08/19/science/dark-spots-in-our-knowledge-of-neptune.html?_r=0
JUNO mission to Jupiter http://www.nasa.gov/mission_pages/juno/overview/index.html