Has AdS/CFT any predictive power in the natural context? Has AdS/CFT any predictive power in the case of the space-time as we know it and in the case of reality as we know it experimentally?
I may add: AdS/CFT in the interpretation given from the perspective of heavy ion collisions has been falsified (I doubt you can say anything different without getting into fitting parameters trouble). So, how does AdS/CFT relate to reality? 
 A: Short answer: Yes, there are predictions derived from holography that are supposed to describe the real world, e.g. the famous $1/4\pi$ predicition of the bound of the shear viscosity over entropy density ratio by Policastro, Son and Starinets. However, at this stage most holographic descriptions do not claim to make exact predictions for real life systems. Holographic models are still used due to the lack of other tools and since qualitative insights can be gained. The hope that more quantitative predictions can be made is very well justified, but much understanding is still missing.
Long answer: First, one has to clarify if AdS/CFT refers only to the claimed duality between type IIB Superstring theory on $\text{AdS}_5\times \text{S}^5$ and $\mathcal{N}=4$ super Yang-Mills theory or if it refers to the much broader field of gauge/gravity dualities in general (as the term AdS/CFT is often used nowadays). I'll say more about both cases below.
A few words to set the stage before addressing predictability. There are two ways in which gauge/gravity dualities could be applicable:


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*A description of a fundamental interaction might be holographic, e.g. quantum gravity could be describable as a lower-dimensional quantum field theory. Would this mean that we live in a universe of a different dimension? Well, as both descriptions are completely equivalent there is no way to distinguish and the question becomes a philosophic one.

*An effective description of some systems can be holographic, e.g. gravitational models for superconductors. In this case, one does not claim that the gravitational description is fundamental. It just captures the organisation of degrees of freedom of some system in a certain regime. This is comparable to chiral perturbation theory being used as an effective description of QCD below the hadronization scale, where the traditional perturbative tools of QCD fail to make predictions but chiral perturbation theory yields useful insights.


First, consider AdS/CFT in the strict sense, i.e. duality between type IIB Superstring theory on $\text{AdS}_5\times \text{S}^5$ and $\mathcal{N}=4$ super Yang-Mills theory. We know from observations, that the universe is not $AdS$, it does not have a negative cosmological constant. Observations point in the opposite direction, our universe seems to be slightly $dS$ with a positive cosmological constant. Moreover, physics in our universe is certainly not conformal as it looks very different on different scales. $\mathcal{N}=4$ super Yang-Mills, however, is a conformal theory. Consequently, it seems very unlikely that the AdS/CFT correspondence in the narrow sense helps us to describe any fundamental theory of nature.
At this point, we can adress your statement about AdS/CFT being falsified in heavy ion collision experiments. Certainly, heavy ion experiments falsify the statement that QCD is actually $\mathcal{N}=4$ super Yang-Mills. However, nobody ever claimed that exact prediction for heavy ion experiments would be expected from the AdS/CFT correspondence (still using AdS/CFT in the narrow sense here). There are two reasons, why AdS/CFT is still used to try to understand heavy ion collisions:


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*We are desperate. Traditional perturbative descriptions of QCD fail to describe many features of heavy ion collisions. There is just no other tool at the moment than AdS/CFT to describe many of the features of the strongly coupled medium that is created in these collisions.

*Even though $\mathcal{N}=4$ super Yang-Mills is not QCD, some qualitative features of the theories are similar. So one hopes to get some insights via the gravitational description using the AdS/CFT correspondence. These are only qualitative insights and the reason for these applications is not only to make predictions about heavy ion experiments, but also to refine our understanding of the correspondence. You can read about recent progress in the application of AdS/CFT to heavy ion collisions in this very nice review by Chesler and van der Schee. A more comprehensive, but less recent introduction is this review.


We want to understand the correspondence better, since AdS/CFT teaches us that the seemingly remote fields of gravity and quantum field theories are somehow deeply connected in a sense that is not completely understood.
As mentioned above, gauge/gravity dualities are a much broader field today that only the AdS/CFT correspondence. The AdS/CFT correspondence is the best understood example of a holographic duality. This is why it is so widely used. The dictionary between bulk and field theory quantities is less well understood for other dualities. However, there are many more dual gauge/gravity dualities. (For example, the discovery of certain Sasaki-Einstein manifolds and their dual quiver gauge theories have provided a countably infinite number of test of the correspondence by rightly predicting the field theory central charges from the gravity side, which was done here.)
If one would be able to find a putative dual gravity theory of QCD, then one would indeed expect quantitatively right predictions. However, such a dual has not been found yet. I choose to be agnostic about the possibility of such a dual being found in the future. But the lack of other available tools certainly justify the ongoing inquiry.
The questions of effective descriptions of certain systems (most prominently in condensed matter physics) via a holographic dual is yet another story. Condensed matter physicists are usually more open towards the holographic approach than particle physicists are, as they are used to working with effective IR models that are a reorganisation of the degrees of freedom of the UV fundamental (?) theory, usually QED. The idea of holographic condensed matter applications is that the degrees of freedom somehow organise as a gravitational theory. This seems very mysterious at first, but it is almost natural given that we already know that some field theories do have gravitational duals. So far, these effective models have provided qualitative insights into several systems, but no quantitative prediction. The difficulty is to find a gravitational dual to a given system, as the usually intuitions about electrons, bands, etc. gets completely scrambled in the gravity language. Therefore, in order to gain quantitative insights, much more needs to be understood.
