The Alpha Magnetic Spectrometer has found is that there are interesting features in the cosmic ray background which could indicate dark matter particle anti particle annihilations.  That is not the same as finding dark matter.   The ANTARES neutrino telescope was used to take observations of the solar neutrino flux with directionality, which found no excess of neutrinos from the sun  (  That would have indicated dark matter concentrated at the center of mass of our solar system.   Taken together these results indicate that many of our assumptions on the nature of dark matter are wrong.

The following basic questions may sound odd, to a long time student or professional of physics, but dark matter has been popularly missold and is widely misunderstood. Allow me to clarify two points for the benefit of all readers.     

Why would Dark matter particles (WIMPs) collect in the sun?  
WIMP's are matter and are assumed to interact with gravitational fields in exactly the same way as ordinary matter.  

This means that wimps would be attracted to the most massive objects in a collection of mass.  On the cosmological scale we are just a bit of ordinary matter residue in a ocean of WIMPs.  On the local scale, in the simplest formulation, the dark matter should be denser near a massive object.    Therefore it would be more likely to annihilate and produce a signal detectable with a neutrino telescope like ANTARES. 

Why would dark matter particles annihilate into positrons, electrons, or neutrinos? 

Dark matter WIMPs are still matter.  Like ordinary matter the basic laws of quantum physics still apply.  These laws dictate that most every particle has an antiparticle.    Hence when they do interact they would annihilate.  

In the simplest formulation, wimps interact by the weak nuclear force and gravity.  So when they annihilate instead of producing photons, as familiar matter would, they would produce particles associated with the weak force.   Which would in turn produce the particles that the AMS and ANTARES have looked for.

The WIMPs are also not seriously thought to be just one kind of particle.  There are likely to be many many kinds of WIMP.   These could be analogues of the particles of ordinary matter (they don't have to be but it's what we have to go on).  

What do the AMS and ANTARES findings say when considered side by side?

ANTARES found, in simple language, not a sign of dark matter - dark antimatter annihilations from a supposed concentration of dark matter inside the sun.  

Which would indicate that dark matter does not collect where there are concentrations of mass for some reason.  This could also indicate that dark matter does not interact by the weak force or it interacts by the weak force and other forces we don't feel.  Say, a force or two more analogous to electromagnetism.  A force which has a larger interaction probability than the weak force.  Yet, because we can't interact with that force we just don't see the result. 

AMS found interesting features in their data which could indicate dark matter - dark antimatter annihilations occurring elsewhere.     They did find that we may want to perform more of these kinds of observations in the future.   

AMS did not "find" dark matter.  They simply did not eliminate it as a possibility on particle physics grounds.  

AMS and ANTARES results taken together indicate that the most widely accepted assumptions about the nature of dark matter need to be seriously questioned.    Right now dark matter, with certain features, is a key part of our theory of the universe.  The CDM in comes from "cold dark matter".    One assumption made by cosmologist has been that dark matter is essentially collisionless.   It does not even interact with itself. 

This data could mean it is truly "collisionless" and almost does not interact with any other matter, even it self, at all, other than by gravity.   Which means we will never really prove it exist once and for all by detecting it in a particle physics experiment of any kind.  The assumption that dark matter is collisionless is a key part of the astrophysical and cosmological understanding of it.    The very famous bullet cluster result depended on the "collisionless" dark matter just passing through as if it did not interact at all.  Then observing gravitational lensing due to that dark matter.  

Observations such as the bullet cluster evidence will explain the data from cosmology and astronomy.  However, if one is interested in astroparticle physics and particle physics that isn't enough.  If dark matter is really truly, utterly, and totally, collisionless we will never really prove it exists.  There will always remain a possibility that a clever or complex field theory of some kind would explain it away....unless we actually detect a particle of it.     I am sure I am not alone in that opinion. After all, there are many teams of astroparticle physicist looking for the results of dark matter anti dark matter collisions and annihilations.   

The other possibility is that some new ideas are needed.

Given the AMS and ANTARES results dark matter either does not interact with itself in a way that anyone has yet predicted, or it does not interact with gravity in the usual way.   Perhaps it's a combination of both of those possibilities.   Then there are all the results which indicate dark matter exist at cosmological distances and at the scale of galaxies and galaxy clusters.   What could explain these results?

I have what I feel is a very good mathematical model for the  framework which could explain these results written up and under peer review.  The model basically predicts that dark matter would be sensitive to the strength of gravity in a way that normal matter isn't.  It will either decay into pure gravitational energy, or annihilate to particles of a field analogous to electromagnetism but which we do not interact with.  Thus it cannot be concentrated near dense clumps of matter.  

We should consider the M-Theory inspired ideas of people like  Lisa Randall, Paul Steinhardt and Niel Turok, that dark matter is simply ordinary matter which lives on nearby M-Theory brane.  The "branes" in M-Theory cosmology can represent essentially separate universes.  Gravity can leak from one brane to another.  Therefore we are feeling the gravity of other universes but not seeing their matter.    Such an idea would gain more traction if supersymmetry could be observed.  

The bottom line:
 That AMS did not find dark matter, only interesting bumps in their data which bear further investigation.  ANTARES did not find dark matter sings from the direction of the sun, not even bumps.  Taken together these indicate that our most basic assumptions about dark matter need to be, at least, reconsidered.  
Updated to add:
It should be noted that this experiment shows the value of human space flight.  The ISS has been criticized as being more of a welfare program which never produced any significant pure science.  This may be every bit as significant as the observations regarding the Higgs Boson, and the CMB.   

Bravo NASA, Bravo.