Are they the same game? Or is D&D Minis like an abridged toned down version of D&D 4.0? Or is it a separate rule-set entirely?
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D&D miniatures is an entirely different game. The main similarities are:
Otherwise they have nothing to do with one another.
At their core the Minatures version is a simplified version exclusively devoted to tabletop war-gaming, rather then Roleplaying. And is pretty much a spiritual successor to the war-game series that D&D eventually spawned from.
HOWEVER, depending on the edition, the manuals would often include rules for exclusive items, classes, spells, etc, that could be used in the Roleplaying version as well. And of course playing D&D with miniatures helps keep better spacial awareness when dealing with combat, or other complicated situations when you needed a guide for where people were standing (and using the actual miniatures was more immersive then just using cheap marker-tokens, especially if you didn't know what a particular creature was supposed to look like).
Pretty much it's just an extra thing for you to do to get a little more value out of your D&D investment. Since you get tired of war-gaming you can use your miniatures to help with immersion for a roleplaying session or vice versa.
WotC does a lot of cross over between the two. You will often, for example, see a Mini's power make it into the Character Builder. So mechanically, they are very similar. Also, the mini's sets are designed to be used in 4ed games in addition to the skirmishing game, so they tend to follow similar release formats. For example, Eberron themed mini's came out around the same time as the Eberron CS was released.