What property (besides charge) of antimatter is different than matter? There must be a property that is different other than charge.  When two oppositely charged matter particles interact, they do not annihilate.  I'm told that two neutrally charged matter/antimatter particles will also annihilate.
Experimentally speaking, if I give you a particle and don't let you measure its charge and also don't do anything to annihilate it, is it possible to tell if that particle is matter or antimatter? Thanks.
 A: All internal quantum numbers are inverted (baryon number, lepton number etc).
When thinking about antimatter, it is important to not think of it as being the 'opposite of matter', but to think of it's defining property as being that it will annihilate with matter to produce energy.
Once we think about it this way it is clear that any quantum numbers, which are conserved in particle interactions, must be inverted in an antiparticle, in order to conserve the number in the overall interaction of annihilation.
To answer the second part of your question, I believe that the identities of particles are typically found by observing their paths in cloud chambers, or in more modern terms, the detectors of particle accelerators.
This process involves looking at the direction that the particle curves in a magnetic field, but also looking at the way the particle decays.
In nature fundamentally, there is a difference seen between matter and antimatter. Beta decay disobeys P-symmetry, meaning that the world would not be symmetrical under a mirror image of itself. This was found in beta decay of cobalt 60, where the emitted electrons were preferentially emitted in the opposite direction as the gamma rays emitted; this meant that most of the electrons were emitted in the direction opposite to the nuclear spin. This showed a violation of P-Symmetry as the universe would not look the same, in this experiment, under a mirror image of itself.
What was found however is that this phenomenon, a weak interaction, was symmetrical under CP-Symmetry, meaning that a universe would appear the same as it's mirror image if all internal quantum numbers were also inverted ( the matter was swapped with antimatter). This shows that on a fundamental level, there is a difference between matter and antimatter, as a swap of internal quantum numbers (antimatter), will also require a reflection of coordinates (right to left hand) in order to preserve symmetry.
Put briefly, the weak interaction is not symmetrical under an inversion of charge (and by charge I mean an inversion of all internal quantum numbers). By this, it follows that our universe would not look the same as one in which all matter was replaced with antimatter, and so, yes, there is a fundamental difference between matter and antimatter.
Up until now, we have thought that all forces are symmetrical under CP-Symmetry, however, there has been some debate about CP violation in the decay of the neutral kaon, however, I feel like I have already gone beyond the scope of the question.
Hope this helps.
A: 
Experimentally speaking, if I give you a particle and don't let you measure its charge and also don't do anything to annihilate it, is it possible to tell if that particle is matter or antimatter?

Yes it is possible. Say you are accelerating your particles hitting them with photons. By this


*

*you could determine wether it is an electron/positron or a proton/antiproton Due to their lower masses in comparison to the proton/antiproton the acceleration of the electron/positron is stronger,

*these subatomic particles receive photons, gain kinetic energy and accumulate photons.


If one apply an external magnetic field to these subatomic particles - perpendicular to the direction of propagation - the particles are deflected in different directions. If the electron/antiproton gets deflected to the left, then the positron/proton gets deflected to the right.
Even if you don't observe these particles you could observe a photon emission, which takes place from these particles with their kinetic energy and gained photons. The deflected electron/antiproton emits photons - in relation to the described case above - to the right and the deflected to the right positron/proton emits photons to the left.

What property (besides charge) of antimatter is different than matter?

It is well known that beside the charge subatomic particles have two more intrinsic properties. This are their intrinsic spins and their magnetic dipole moments. Wherein both are unidirectionally connected, the spin of a particle and the magnetic dipole moment determine each other. If for a particle the magnetic dipole moment and the spin are pointing in the same direction then for the same antiparticle this properties always are pointing in opposite directions.
Perhaps of which a rule could be derived: Two particles of different charges would annihilate only if the spin-magnetic dipole-orientations are different for these particles. Which for example is the case for the electron and positron, but not for electron and proton interaction.
