I don’t think it is correct to say that there are three versions of the Vulgate. I’m not use how one would even count the number of versions or editions.
There is the Latin Bible prior to Jerome, of which little is extant, called the Vetus Latina. This was not one version, but many different variant manuscripts. It is not generally called ‘Vulgate’ but is its precursor.
Jerome’s Vulgate is not extant in its entirety, but only in numerous manuscripts which have been edited much since the time of Jerome. Jerome’s work with the Bible was complex. He translated some books, and merely edited others. He sometimes worked from Latin source texts, and other times from Greek (and later Hebrew) texts.
There are a number of different Vulgate editions of the Bible between the time of Jerome and that of Pope Sixtus V.
Pope Sixtus V completed (in 1590) work begun under earlier Popes, spurred by the Council of Trent, of a supposedly definitive edition of the Latin Vulgate Bible. Over 6,000 manuscripts were consulted for this 1590 edition of the Vulgate. Pope Sixtus V forbid the text to be changed. But only a few years later, Pope Clement VIII amended the 1590 edition, publishing three slightly different editions, in 1592, 1593, 1598. This later edition is generally referred to as the Clementine Vulgate.
There have been a number of prominent edits of the Clementine Vulgate, such as the 1861 Vercellone edition and the Hetzenauer editions from the early 1900’s. Leander van Ess preserved all four Sixtus V and Clement VIII editions in his edition of the Latin Vulgate which notes the differences between the editions in various footnotes.
The edition linked to above at sourceforge is the 2006 Tweedale edition, a lightly edited edition of the Clementine Vulgate.
The Neo-Vulgate is not really in the Scriptural tradition of the Vulgate. It follows the Hebrew and Greek textual variations more often than the Latin ones. It reads more like a loose paraphrase translation of the Hebrew and Greek into Latin, rather than a critical edition of the various Latin source texts.
Ron