What is the difference between a magnon and a spinon?

The two answers given so far are wrong.

A magnon is an excitation carrying spin-$1$. A spinon is an excitation carryong spin-$\frac{1}{2}$. This has nothing to do with it being an excitation above a ferro- or antiferromagnet. The difference is much more dramatic, such that magnons are 'normal'/standard, yet spinons are very special.

Suppose you have some $SU(2)$-symmetric Hamiltonian (in more than one spatial dimension) made out of spin-$\frac{1}{2}$ particles which is in a ground state that spontaneously breaks the $SU(2)$. If you flip a single spin you've created a magnon, not a spinon. Intuitively you might think a spin-flip in a spin-$\frac{1}{2}$ system carries a half-integer spin, but that's not true: while it is true that $|\downarrow\rangle$ and $|\uparrow\rangle$ each carry a half-integer spin, their difference is an integer spin. In other words: magnetic phases (in more than one dimension) have magnon quasi-particles.

Spinons are much weirder. In fact since any local spin operator changes an integer amount of spin, you cannot create a single spinon with a local operator! Hence spinons are examples of fractionalized particles: they can only arise as part of a physical disturbance. For example spin liquids can give rise to spinons. A less exotic but still nice example is the spin-$\frac{1}{2}$ Heisenberg chain in one dimension where a single spin flip actually creates two emergent quasi-particles (two spinons). In a way a similar thing already happens in the transverse-field Ising chain $H = -\sum S^x_n S^x_{n+1} + g \; S^z_n$: imagine going to the ordered phase and applying a single spin-flip operator -- can you see how this actually creates two quasi-particles?