Let me start by saying that I understand the definitions of the Lie and covariant derivatives, and their fundamental differences (at least I think I do). However, when learning about Killing vectors I discovered I don't really have an intuitive understanding of the situations in which each one applies, and when to use one over the other.
An important property of a Killing vector $\xi$ (which can even be considered the definition) is that $\mathcal{L}_\xi\, g = 0$, where $g$ is the metric tensor and $\mathcal{L}$ is the lie derivative. This says, in a way, that the metric doesn't change in the direction of $\xi$, which is a notion that makes sense. However, if you had asked me how to represent the idea that the metric doesn't change in the direction of $\xi$, I would have gone with $\nabla_\xi g = 0$ (where $\nabla$ is the covariant derivative), since as far as I know the covariant derivative is, in general relativity, the way to generalize ordinary derivatives to curved spaces.
But of course that cannot be it, since in general relativity we use the Levi-Civita connection and so $\nabla g = 0$. It would seem that $\mathcal{L}_\xi\, g = 0$ is be the only way to say that the directional derivative of $g$ vanishes. Why is this? If I didn't know that $\nabla g = 0$, would there be any way for me to intuitively guess that "$g$ doesn't change in the direction of $\xi$" should be expressed with the Lie derivative? Also, the Lie derivative is not just a directional derivative since the vector $\xi$ gets differentiated too. Is this of any consequence here?