🚀 go-pugleaf

I have been trying to learn General Relativity by reading Robert Wald's 
book and have become confused about the difference between a
covariant derivative contracted with a vector and a Lie
derivative in the direction of the vector.  Any help clarifying
the distinction would be appreciated. Specifically:

If D_a (ie, D sub a) is the unique derivative operator induced by
the metric which has the property D_a g_bc = 0  (page 35) then the
derivative of a Tensor (T^b1b1..._c1c2...) when transported in a 
direction t^a (ie, t superscipt a) is  t^a D_a T^b1b2..._c1c2 ...

But the definition of Lie derivative of a tensor with respect
to a vector v^a is defined on page 439 as the limit of the
diffence of the tensor and its pullback divided by t, the
parameterization of the curve along which T is being moved
(v^a is the tangent to the curve at the current position).
How is this different from v^a D_a T ?  Wald clearly
presents it as different since on page 441 (eq. C.2.13)
he shows that L_v u_a  (ie, the Lie derivative of the dual vector
u_a in the direction v^b) = v^b D_b u_a + u_b D_a v^b .
If the Lie derivate were equal to the covariant derivative
then the second term would not be there.

So, conceptually, what is the difference between the two?
Does one require a different type of topogical space or
parameterization?  Mathematically how do they differ?
thanks in advance.

			Larry.