Ignoring the different fullscreen/widescreen releases, there's basically been three separate releases of E.T. on DVD.
Back in 2002, Universal released two different versions: A two-disc set, which contained both the original 1982 cut and the 2002 20th Anniversary Special Edition cut, and a four-disc Ultimate Gift Set, which contained the 2002 cut on the first disc, the 1982 cut on the second, a third disc of bonus features, and the fourth disc, which was the soundtrack CD. It also contained a nice hardcover making-of book, as well as a certificate of authenticity and a film frame. All of the versions released in 2002 are now out of print.


Last year, Universal re-released E.T. as a one-disc set--and only as the 2002 Special Edition cut, dropping the original 1982 version.

Here's a DVDTalk thread which covered the new single-disc release:
Universal dumbs down for re-releases of E.T., Cape Fear, and Meaning of Life 08.23.05From that thread:
Universal has just sent us some interesting news regarding the upcoming single-disc re-release of E.T.: The Extra-Terrestrial. Even though this edition shaves off some of the better extras from the two-disc set, including the full-length documentary, it will add an isolated track featuring John Williams' famous score. This track is exclusive to this release. The rest of the disc features the 2002 cut of the film in 1.85 anamorphic widescreen, Dolby Digital 5.1 EX and DTS ES Surround tracks, "The Reunion: E.T.'s Family and Friends" and "Behind-the-Scenes with Composer John Williams" featurettes, E.T.'s Interactive Space Tour, the trailer, production photos and concept drawings. Look for this E.T. on August 23rd with a retail $19.98.
So, basically, stay away from the current version which is selling at Target and Amazon, and probably pretty much everywhere else. With that, you lose the 1982 version, and only gain the isolated score track. (There are separate widescreen and fullscreen releases for this disc, by the way.)
The 2002 releases are far superior--try and find those if you can. Preferably, the Ultimate Gift Set, which may be quite hard-to-find and pricey these days, but contains the most bonus features. (This was released in widescreen only.) The two-disc set should be a little easier to find, and much cheaper than the box. (The two-disc sets had separate fullscreen and widescreen releases, as well.)