The idea of topological quantum computing was introduced by Kitaev in this paper. The basic idea is to build a quantum computer using the properties of exotic types of particles, known as anyons.
There are two main properties of anyons that would make them great for this purpose. One is what happens when you use them to create composite particles, a process we call fusion. Let's take the so called Ising anyons (also known as Majoranas) as an example. If you bring two of these particles together, it is could be that they will annihilate. But it could also be that they become a fermion.
There are some cases where you will know which will happen. If the Ising anyons just pair created from the vacuum, you know that they'll go back to the vacuum when combined. If you just split a fermion into two Ising anyons, they'll go back to being that fermion. But if two Ising anyons meet for the first time, the result of their combination will be completely random.
All these possibilities must be kept track of somehow. That is done by means of a Hilbert space, known as the fusion space. But the nature of a many-anyon Hilbert space is very different to that of many spin qubits, or superconducting qubits, etc. The fusion space doesn't describe any internal degrees of freedom of the particles themselves. You can prod and poke the anyons all you like, you won't learn anything about the state within this space. It only describes how the anyons relate to each other by fusion. So keep the anyons far apart, and decoherence will find it very hard to break into this Hilbert space and disturb any state you have stored there. This makes it a perfect place to store qubits.
The other useful property of anyons is braiding. This describes what happens when you move them around each other. Even if they don't come close to each other in any way, these trajectories can affect the results of fusion. For example, if two Ising anyons were destined to annihilate, but another Ising anyon passes between them before they fuse, they will turn into a fermion instead. Even if there was half the universe between them all when it passed by, somehow they still know. This allows us to perform gates on the qubits stored in the fusion space. The effect of these gates only depends on the topology of the paths that the anyons take around each other, rather than any small details. So they too are less prone to errors than gates performed on other types of qubit.
These properties give topological quantum computing a built in protection that is similar to quantum error correction. Like QEC, information is spread out so that it cannot be easily disturbed by local errors. Like QEC, local errors leave a trace (like moving anyons a little, or creating a new pair of anyons from the vacuum). By detecting this, you can easily clean up. So qubits built from anyons could have much less noise than ones built from other physical systems.
The big problem is that anyons don't exist. Their properties are mathematically inconsistent in any universe with three or more spatial dimensions, like the one we happen to live in.
Fortunately, we can try to trick them into existing. Certain materials, for example, have localized excitations that behave as is they were particles. These are known as quasiparticles. With a 2D material in a sufficiently exotic phase of matter, these quasiparticles can behave as anyons. Kitaev's original paper proposed some toy models of such materials.
Also, quantum error correcting codes based on 2D lattices can also play host to anyons. In the well known surface code, errors cause pairs of anyons to be created from the vacuum. To correct the errors you must find the pairs and reannihilate them. Though these anyons are too simple to have a fusion space, we can create defects in the codes that can also be moved around like particles. These are sufficient to store qubits, and basic gates can be performed by braiding the defects.
Superconducting nanowires can also be created with so-called Majorana zero modes at the end points. Braiding these is not so easy: wires are inherently 1D objects, which doesn't give a lot of room for movement. But it can nevertheless be done by creating certain junctions. And when it is done, we find that they behave like Ising anyons (or at least, so theory predicts). Because of this, there is a big push at the moment to provide strong experimental evidence that these can indeed be used as qubits, and that they can be braided to perform gates. Here is a paper on the issue that is hot off the press.
After that broad intro, I should get on with answering your actual question. Topological quantum computation concerns any implementation of quantum computation that, at a high level, can be interpreted in terms of anyons.
This includes the use of the surface code, which is currently regarded as the most mainstream method for how a fault-tolerant circuit model based quantum computer can be built. So for this case, the answer to "How do Topological Quantum Computers differ from others models of quantum computation?" is that it doesn't differ at all. It is the same thing!
Topological quantum computation also includes Majoranas, which is the route that Microsoft are betting on. Essentially this will just use pairs of Majoranas as qubits, and braiding for basic gates. The difference between this superconducting qubits is little more than that between superconducting qubits and trapped ion qubits: it is just details of the hardware implementation. The hope is that Majorana qubits will be significantly less noisy, but that remains to be seen.
Topological quantum computation also includes much more abstract models of computing. If we figure out a way to realize Fibonacci anyons, for example we'll have a fusion space that cannot be so easily carved up into qubits. Finding the best ways to turn our programs into the braiding of anyons becomes a lot harder (see this paper, as an example). This is the kind of topological quantum computer that would be most different to standard methods. But if anyons can really be realized with very low noise, as promised, it would be well worth the small overheads required to use Fibonacci anyons to simulate the standard gate based approach.
