Since the second two are Adobe I will assume you are referring to Adobe versions. There might be some slight differences contour shapes between the early Adobe versions and the Pro and Premium versions. The main difference is the number of extras that the newer versions contain. Premium was included with the CS3 version of creative suite and has a very large number of glyphs including a lot of ornaments. The Pro version is a bit smaller than the Premium and that is what now ships with CS4.
Yikes, this is completely wrong. The two Adobe families mentioned in the OP are Garamond Premier Pro and Adobe Garamond Pro. Adobe Garamond was one of the first Adobe Originals, designed by Robert Slimbach and released in 1989. It was later converted to OpenType and renamed Adobe Garamond Pro.
Adobe Garamond Regular
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Adobe Garamond Pro® - Desktop font « MyFonts. An Adobe Originals design, and Adobe’s first historical revival, Adobe Garamond is a digital interpretation of the roman types of Claude Garamond and the italic.
Garamond Truetype Font
Garamond Premier Pro is a completely separate Garamond revival, also by Robert Slimbach. It is less regularized (Adobe Garamond had to be designed to accommodate less powerful software and hardware of its day) and has four optical sizes. Garamond Premier Pro also has Greek and Cyrillic.
Monotype Garamond Pro Bold Font
There is no 'Premium' version. They both support CE languages, so they both have the 'Pro' suffix.