I have owned the Criterion edition of this film a long time; I recently bought the Ruscico (Russian Cinema Council) edition and think I should try to make clearer the differences. This is not a critique of Tarkovsky's work -- that is beyond my capabilities.
As you may already know, the Criterion edition is taken from Martin Scorsese's personal print and represents the penultimate version of the film, while the Ruscico edition represents the release version, which is about twenty minutes shorter. However, Tarkovsky did more than pare twenty minutes off the film -- it's actually a somewhat different film, though the differences are not major.
To begin with, the Scorsese print (Criterion) has a completely different set of credit titles and intertitles, and at that stage the film was titled "Strasty po Andreyu" (Passion of Andrei). The release version (Ruscico) is titled "Andrei Rublev" and is not merely shorter: it contains shots that do not appear in "Strasty po Andreyu" (Criterion). Commenting on the DVDs themselves, the Ruscico DVD is much better looking. The subtitles (as one might imagine) are written by someone whose native language is Russian, and that is very important to me. When the subtitles are written by an English-speaker they are rendered in English idioms and subtle, specific meanings are often lost. Sometimes one cannot even tell what a scene is about. (There is a scene in Criterion's "Ivanovo Detstvo", for example, where the English-written subs completely obscure the point of a scene, while the Russian-written English subs in Ruscico's version make it perfectly clear.) Russian-written English subtitles are sometimes ungrammatical, use idioms whose meanings are unclear to non-Russian speakers, and sometimes even inadvertantly use a word from yet another language (French, in one case that I saw), but I'll take subs written by someone whose native language is that of the film any day. In fact, if you see a version of any foreign film with English-written subs first, then see a version where the English subs were written in the film's country of origin, it will be like seeing a whole new film. (A spectacular example is the difference between Kino's "Zerkalo" [Mirror] and Ruscico's -- there is NO comparison [Ruscico wins!], except that you have to avoid Ruscico's 5.1 audio remix and select the original mono.)
Additionally, an extra of great interest is hidden away on Ruscico's "Andrei Rublev" DVD. In the individual filmographies certain titles are highlighted: these are accompanied by trailers, three of which are for Tarkovsky films. These trailers are made up largely of shots that are entirely different from anything that appeared in the final film, so should be of absorbing interest to any fan of his work.
To sum up: Although I prefer the long version represented on Criterion's disk ("Strasty po Andreyu"), the Ruscico disk has a superior image, better subtitles (to my way of thinking), and fascinating extras if you can find them. Get both DVDs.