Physicist: Anti-matter is exactly the same as matter, but different. If you, and everything else on the planet, were suddenly turned into anti-matter, you’d never know the difference. While the “anti-” of anti-matter may seem to give it an air of mystery, it still acts just like ordinary matter in essentially every respect. Specifically, anti-matter carries positive energy and mass, just like regular matter, while negative matter carries negative energy and mass.
Famously, when you bring matter and anti-matter together they annihilate. All of their combined mass is converted into buckets of energy in an amount dictated by Einstein’s little-known equation, E=mc2. For comparison, the largest nuclear device ever detonated, the USSR’s “Tsar Bomba”, is the yield you’d expect from about 1 kg of anti-matter (so about 2 kg of energy total, because it needs some mass to annihilate with). A single atom of anti-matter (say, anti-carbon) annihilating in your ear would be just barely audible as a pop. It’s the energetic equivalent of a ant stomping a foot (in anger!).
Negative matter, more commonly called “exotic matter”, has negative energy. If you were to bring it into contact with ordinary matter you would see, not an awesome explosion, but an underwhelming and abrupt nothing. When exotic matter in brought together with ordinary matter, the positive energy of the matter and the negative energy of the exotic matter cancel out entirely, leaving nothing at all behind.
Exotic matter is a freaking blank check for sci-fi writers. Warp drives, worm holes, perpetual motion, and even time machines are possible if you allow for negative matter and energy. In fact, Hawking proved that if you want to build a time machine smaller than the universe, negative energy/mass is a requirement. Spacetime, as described by general relativity, is pretty limited by the fact that energy and matter seem to be strictly positive. About the weirdest things you’ll see are black holes, which are pretty cool, but… time machines. With a liberal peppering of exotic matter (often far more than the universe’s total stockpile of regular matter) you can really open up the flood gates of the weird.
However the big difference, arguably the biggest difference, between anti-matter and negative matter is that negative matter doesn’t exist.
There are some subtle physical laws that imply that the creation of negative energy, in the form of exotic matter or not, has limitations called “quantum interest“. Anytime a bit of negative energy is generated (and the methods involved create, like, none), a larger, overwhelming pulse of positive energy must be created almost immediately. In fact, we’ve never directly observed negative energy and it’s very, very likely that we’ll never be able to do more than infer that negative energy exists.
But anti-matter definitely exists, and can be created and stored (a few particles at a time) here on Earth. Many particle accelerators today generate and use anti-protons all the time. When you smash stuff together, or otherwise get a mess of energy in one place, new particles are generated; half matter and half anti-matter. It’s basically annihilation in reverse. Once you create a spray of new particles, you sort the matter and anti-matter apart, keep the anti-particles ionized, and store them (briefly) in a “magnetic bottle“. If they ever becomes electrically neutral the magnetic bottle stops working, and they fall and annihilate with the ordinary matter at the bottom of the container. Anti-particles are totally the hot potatoes of particle physics.