In terms of scientific and engineering challenges, which one is closer to fruition? Specifically I am looking for the feasibility of using one of these systems for space-only travel between the orbits of planets and bodies throughout our solar system. Bonus points if the propulsion system can maintain constant acceleration, like the ships in The Expanse, and provide crew members with some fraction of Earth gravity.
Antimatter is a battery. Fusion is a generator. They're good for different things.
Unless we find a way to mine antimatter from some source we don't yet know, it will only ever be stored energy, made with planetary particle accelerators (or some future alternative), and every gram made will require input of its mass equivalent in energy.
Fusion, on the other hand, has at least the potential for portable power systems and engines, likely even capable of refueling "in the field", either from dense nebulae, or from planetary atmospheres and oceans.
Antimatter has a far higher energy density and release rate -- fusion has a much longer duration of action. It's like the difference between gunpowder and gasoline, only far more so. For some things, you need lots of energy, all at once -- that's a potential application for antimatter. For other things, you need your energy to be delivered over time -- that's a better place for fusion.
Fusion power. In fact, we (probably) have done all of the basic science that's needed to build fusion power plants, the only thing that remains is a very expensive development process.
The process of fusion is pretty well understood: you cram a bunch of very hot hydrogen together, it fuses into helium, and releases a bunch more energy. You use that energy to boil steam, and after that a fusion power plant looks just like a regular coal, gas, or nuclear power plant.
The problems are technical, not conceptual:
How do you reliably start a "cold" fusion reactor by cramming enough hydrogen together and getting it hot?
Once the fusion reaction is started, it generates so much heat that any fusion reaction naturally wants to blow itself apart. So how do we keep that fusion reaction stable enough that we can keep feeding it new hydrogen and keep the reaction going?
These are really questions of the geometry of the reactor, of the mechanisms that need to be built, of how many and how strong of magnets do we need to confine the reaction, etc. We will have to do a lot of very expensive modeling and simulation, then build prototypes, then go back to the design stage and improve those prototypes, then build more prototypes. Eventually we get to the point where all of the apparent kinks are ironed out and we will build the world's first pilot fusion plant. That will be another big learning experience, and probably 5-10 years after that you will see the first generation of Mark 1 fusion reactors starting to be installed around the world.
The only problem is that all of this is terribly expensive. Unless there is a clear and convincing reason to spend billions to trillions of dollars developing this technology, it's much cheaper not to. From a technical standpoint nuclear power is just as clean as fusion power would be, and in the modern era there's much more of a focus on making renewables incrementally cheaper and more efficient rather than building a whole new technology. (Note that the problem of nuclear waste and nuclear contamination are more political problems than technical problems.)
This is what caused one of the early fusion pioneers, Lev Artsimovich, to say, "Fusion will be ready when society needs it."
The science is done, but the very expensive development has never been funded. Fusion has been "20-30 years away" since the 1950's because we got to the point where we can't go further without massive investment, and that investment just hasn't happened.
Fusion, but probably with an antimatter boost.
Simply put, we have been working on it for decades and are already on our way to having it.
It has already been noted in other answer's and in comments that one of the biggest differences is that fusion is much better for sustained energy release, while antimatter releases more energy immediately. A purely antimatter drive would probably be more akin to early designs of exploding nukes behind a rocket, except it would be detonating antimatter (in layman's terms). Likewise, our current ability to produce antimatter is heavily limited by energy production and confinement (described as a magnetic bathtub in a documentary I saw once), which will likely be supplemented by the completion of fusion reactor designs.
I would like to note that fusion reactors and fusion rockets are two different things, but it stands to reason that once you have reactors, rockets aren't too far behind. Personally, I am much more convinced that something between spiked fusion rockets and antimatter-catalyzed nuclear pulse propulsion are much more likely to become the standard.
Feel free to correct me if I'm misunderstanding literal rocket science.
Basically, fusion and/or fission rockets are the primary propulsion method, however antimatter is used to essentially supercharge the entire reaction. This works differently for fusion vs fission, but in either case the end result is much greater impulse and fuel economy from the drive than you would get with either method on it's own.
For inertial confinement fusion it would save the initial energy investment necessary to start the reaction. Some design variants speculate much higher plasma gain and/or impulse.
For a fission rocket using the nuclear pulse propulsion design (back to firing nukes behind the rocket) it would essentially give you more bang for your buck. The comparison as I understand it is essentially turning your gen-1 A-bomb into a thermonuclear H-bomb (grossly over simplified).
Rockets which use antimatter as their primary source of energy are perhaps easier to conceive of and make than most fusion rocket designs (because you just throw some antiparticles at your reaction mass, and blow the resulting kaboom out of the back of your ship via a rocket nozzle, probably a magentic one, job done) but the sheer unimaginable cost of producing and storing the damn stuff in bulk will stop any such plans dead.
By "bulk" I mean "milligrams" here, by the way. Getting lkarge amounts is going to require vast infrastructure built in space, which will be the end product of a mature spacefaring civilsation, and such a civilisation will have need of decent rockets. Something will have to come first.
The easiest solution is of course the good old Orion Drive, which offers good efficiency and high thrust-to-weigh, and might well be possible with modern day technology. Most of the boom of modern nuclear weapons is provided by fusion reactions, after all. Job done!
This isn't an entirely smooth thrust, of course, but non-pulsed nuclear engines are likely to be exceedingly hard to make... witness our inability to make net-positive-output, non-pulsed nuclear reactions on Earth, and getting anything of that sort of work will be much easier than harnessing it as a powerful rocket of the sort that will shove you around the solar system. No; your best bet will be some kind of pulsed nuclear propulsion... designs like the Magneto-Inertial Fusion or Magnetized Target Fusion seem relatively plausible, and both are pulsed, for example.
Bonus points if the propulsion system can maintain constant acceleration, like the ships in The Expanse, and provide crew members with some fraction of Earth gravity.
As a footnote, be aware that brachistochrone transits may be fast, but they require very, very high performance engines. Centrifugal gravity combined with much more modest boost-coast-brake transits will be vastly easier, and allow for clever external beamed propulsion systems to maximise efficiency and leave the heaviest and most inconvenient parts of your rocket at home.