The incomparable PZ Myers cites an empirical study purporting to show that clones could not be biologically identical. This suggests a couple of interesting philosophical points to me.
First is about what the study actually shows. The study is of identical twins, who are pretty close to being genetic duplicates, and tries to draw from them some conclusions about actual genetic duplicates. It's a nice little bit of inductive reasoning, and it seems like a reasonably sound conclusion to me. Of note, though, is the effect that environmental features had on the development of the twins, features which might not affect the development of a clone. In other words, if it were possible (logically, it is; physically, it may not be) to exactly duplicate the conditions under which oneself grew up, could one's clone become an exact duplicate of one's past self? It seems possible. But this then suggests that the issue of identity is not going to be settled neatly by science; we need to roam into the realm of philosophical analysis to really settle the question.
This leads to point two, namely about the notion of identity itself. I'm going to pick up on a pair of comic book characters, Cable and Stryfe. The backstory is appropriate complex, so I'll just focus on the relevant detail, namely that Stryfe was Cable's clone, cloned such that he was exactly biologically identical to Cable, and then both were raised in entirely different environments. The question is: were Stryfe and Cable ever strictly identical?
That is, we can say things are "identical" if we mean they are "really, really similar, so similar they cannot be easily distinguished". Call that "lax identity". We can also say things are "identical" if we mean that they are actually indistinguishable -- even an omnipotent, omniscient intellect would not be able to tell them apart. Call that "strict identity". Clearly, Cable and Stryfe were laxly identical. Just as clearly, they were not strictly identical: for one, their bodies would not be made up of the same basic physical components (cells, for example), but of functional equivalents (that is, cells that would perform the same biological functions in each organism).
But, suppose we warp the story, and say that even this were overcome: even the very most basic particles that made up their bodies could not be distinguished from each other, and their minds (assuming these are not physical) were also not distinguishable in any way. Have I just described an impossibility? Is there nothing that satisfies the criterion of strict identity?
On the face of it, it would be extremely strange if nothing did satisfy the criterion, for it contains no obvious contradiction or paradox (compare, for example, to the Russell Paradox). But there may be a buried one in some of the concepts that are deployed in the criterion.
There is a law of logic known as Leibniz's Law, which states that any two identical things are indiscernible from each other (i.e., the indiscernibility of identicals). There is another claim, which is only a putative law, also (confusingly) called Leibniz's Law, which states that any two indiscernible things are identical to each other (i.e., the identity of indiscernibles). The controversy arises over just the sort of issue I just got to, namely the question of whether there are any two things that are strictly identical. If there are, then it is not a law of logic that two indiscrenible things are identical to each other -- for there would be two (separate, non-identical) things which could not be distinguished from each other.
The usual way to claim that there cannot be any two such things is to assert the identity of indiscernibles, through the claim that any two things must always occupy different locations in space and/or time, and hence can always be distinguished on this ground.
This brings me to the end of my musings, as I still know very little about the philosophy of space and time, and very little about how I could figure out what a location in space and time actually is. I will say this, though: a solution like Kant's seems to me to imply that the identity of indiscernibles is true. Kant claimed, very approximately, that we experience things in space and time only because these are the forms of the most basic mental faculty we possess (called "sense"), the faculty that assembles the random impressions ("sensations") that assail our minds into "intuitions", coherent wholes that can then be used as raw material for the next faculty ("understanding"), which produces thoughts. So, space and time are only orders -- systems of organization, if you like -- in which sensations are placed. But this then suggests that location is relational -- that is, some object A is only at space/time location (x, y) because my mind put it (more strictly, constructed an appearance) there. Since location is relative to an observer (a third object), any two objects that are both not identical to the observer can always be distinguished on the basis of their location.
However, we got into this mess talking about clones, and clones are always of people. So, what of my clone? Can my clone be indiscernible from me? The answer is still "no", under Kant's framework. For, my location as an object in space and time is not fixed relative to myself as an object in space and time, but myself as a consciousness, and the location of my clone in space and time is similarly fixed relative to myself as a consciousness.
What about (and here we get way into science fiction territory) the clone of my consciousness? Well, Kant says that we can't perceive other consciousnesses, so this sort of clone would be imperceptible unless it were identical to or contained by (or some such) a clone of the first type. On the other hand, if we presume that we can perceive other consciousnesses, we still don't have a problem. For, I perceive my own consciousness' location in space and time relative to my own consciousness -- that is, I always perceive its location to be (0,0) -- while the clone of my consciousness would always be located at some other, non-zero, point relative to my consciousness.
Therefore, if we accept the Kantian view of space and time, there cannot be any strict identicals; hence, clones cannot be strictly identical to each other.
Why should we accept Kant's view here? As I said, I don't know much about the philosophy of space and time, so I don't know what else is out there to combat or contrast it with.