According to my book (Hsu: ODE), a solution $\phi(t)$ to the system $x' = f(x)$ that is bounded for all $t \geq 0$ satisfies one of:
1) $\omega(\phi)$ contains an equilibrium, or
2) either $\phi(t)$ is periodic or $\omega(\phi)$ is a periodic orbit.
My question is: what is the difference between the solution being periodic and the limit set $\omega(\phi)$ being a periodic orbit?
As explained in comments, a non-periodic solution whose limit set is a periodic orbit is something spiraling toward a periodic orbit (either from the inside or from outside). An example of such a system is $$\begin{split} x' &= x\cos(x^2+y^2) - y \\ y' &= y\cos(x^2+y^2)+x\end{split}$$ The plot below (courtesy of Wolfram Alpha) shows two periodic orbits, one of which is attracting (and is the limit set for many non-periodic solutions) and the other one is repelling.
This system has infinitely many periodic orbits which are concentric circles, alternating between attracting and repelling.