Q:

How is antimatter distinct from negative mass?

How is antimatter distinct from negative mass? I'm very interested in the concept of negative mass from a strictly mathematical perspective.  (i.e. this question should not be taken as relating to suppositions involving the harnessing anti-gravity.)  Rather, I am interested in the idea of "the opposite of something".  
My understanding is that anti-matter, although having opposite charge to normal matter, still has mass.  Is this correct?
How does negative mass work conceptually?  Is it related to "dark energy" or "dark matter".  Is is purely theoretical?

A:

At the moment, with a quantum theoretical standard model of particle physics where gravity is not included,  antimatter is defined as particles that have all the quantum numbers defining them with opposite sign, so that when particle hits antiparticle all quantum numbers add to zero.
Mass is not a quantum number and is always positive for individual particles, so the annihilation of particle on antiparticle gives energy which can split into completely different particle antiparticle pairs . The cleanest experiments were the LEP e+e- experiments at CERN and other laboratories, where the standard model has been very well validated.
As your link discusses, there are proposals where a negative mass may arise in General relativity but 

General relativity describes gravity and the laws of motion for both positive and negative energy particles, hence negative mass, but does not include the other fundamental forces. On the other hand, the Standard Model describes elementary particles and the other fundamental forces, but it does not include gravity. A unified theory that explicitly includes gravity along with the other fundamental forces may be needed for a better understanding of the concept of negative mass.

So antimatter in the standard model is well validated by the data and mathematically modeled and understood. Negative mass is an open research question.
Part of this research is the coupling of antimatter particles with gravity, and there are experiments studying this at CERN, with anti hydrogen and how it is affected by the earth's gravity.

