Let $T$ be a tensor field on the manifold $M$, $\nabla$ a connection, $v$ a tangent vector at $x\in M$, and $V$ a vector field such that $V(x)=v$.

Then the intuition is as follows:

> The covariant derivative $\nabla_v T$ is the derivative of $T$ **along a geodesic** arc $\gamma$ for $\nabla$ which has direction $v$ at $x=\gamma(0)$.
The derivation is computed in the finite dimensional tangent space $T_xM$, as nearby values $T(y)$, $y\in M$, are compared via parallel transport. 

(Remark: here "geodesic arc" should be made more precise, as geodesics emanating from $x$ are determined as *parametrized* curves and it may happen that the geodesic in the direction $v$ doesn't have velocity $v$)

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> The (value at the point $x$ of the) Lie derivative $\mathcal{L}_VT$ is the derivative of $T$ **along the flowline** of $V$ (passing through $x$).
The derivation is computed in the finite dimensional tangent space $T_xM$, as nearby values $T(y)$, $y\in M$, are compared via pullback along the local flow of $V$.