So let's say I have As Foretold on the battlefield with no counters of any kind on it, and for whatever reason I decide to play Kite Shield with it. I can technically do this, it's just really stupid. Fine, right?
So can I play Ancestral Vision in the same way? I read somewhere that AV counts as CMC 0 for the purposes of tutoring effects; does it count in this one?
Ancestral Visions always has a CMC of 0. CMC is just derived from the Mana Cost. In the case of Ancestral Vision, since it has no mana cost, its CMC is 0. The main difference here is that a mana cost of 0 (which is what Kite Shield has) can be paid. A nonexistent mana cost (which is what Ancestral Vision has) cannot be paid.
However, when you are casting AV off As Foretold, you are not casting it using its (nonexistent) mana cost; instead, you are casting it for a totally different cost. This is referred to as an Alternate Cost. In this case, the Alternate Cost is 0 because that is what As Foretold sets it to.
So, to answer your title question: yes, there is a very important distinction between spells that cost 0 and spells that have no mana cost. However, in the case of As Foretold, that difference often is irrelevant because you are not using the mana cost (or lack thereof) on the spell to cast it.
So can I play Ancestral Vision in the same way? I read somewhere that AV counts as CMC 0 for the purposes of tutoring effects; does it count in this one?
However, when you are casting AV off As Foretold, you are not casting it using its (nonexistent) mana cost; instead, you are casting it for a totally different cost. This is referred to as an Alternate Cost. In this case, the Alternate Cost is 0 because that is what As Foretold sets it to.
So, to answer your title question: yes, there is a very important distinction between spells that cost 0 and spells that have no mana cost. However, in the case of As Foretold, that difference often is irrelevant because you are not using the mana cost (or lack thereof) on the spell to cast it.