A Game of the Year edition was released on September 27, 2011. It
includes all four of the "Cradle of Civilization" map packs, the
"Explorers Map Pack", as well as most of the new civilizations
(Babylon, Spain, Inca, Polynesia, Mongolia, and Denmark), their
respective scenarios, and the official digital soundtrack. However,
"Civilization and Scenario Pack: Korea" and "Wonders of the Ancient
World Scenario Pack" are not included.
A Gold edition was released on February 12, 2013. It includes all
"Cradle of Civilization" map packs, the "Explorers Map Pack", the
"Wonders of the Ancient World Scenario Pack", all the DLC
civilizations and the Gods and Kings expansion pack.
So the Gold edition contains all of the GOTY content, plus the Korea and Ancient Wonders DLC packs and the first expansion Gods & Kings. G&K is the most significant difference between the two editions because it's a full expansion that adds a lot of new mechanics and content to the base game: religions, espionage, new techs, improved combat and diplomacy, plus several more civs and city-states.
Most of the DLC packs on the other hand are extra maps and civs that just give you new options to choose from, but as someone pointed out they are often required to join online games. They are not strictly necessary, especially for single player, but they do offer more variety.
I disagree with the other answer's conclusion; G&K is a massive upgrade that is well worth buying Gold over GOTY.
To address your edit, Gold edition does not include future DLC: there is now a second (and very significant) expansion Brave New World which is not included in either of these editions. To get the full package you would need to have Gold + BNW, or at least Civ5 + G&K + BNW.
Update: The Complete edition has since been released, containing "both expansions and all the DLC packs."