What is the difference between linear space and a subspace? If W is a subspace, is it also a linear space? If V is a linear space, is it also a subspace? I am having trouble wrapping my head around the difference between the two, as it seems that the way the book defines them is the following: both have to have a zero (neutral) element, both are closed under addition and scalar multiplication. Thanks!

I think some of your confusion my arise from being a bit imprecise in what you are saying. One generally says W is a subspace of V. To mean W is a subset of V containing 0 which is closed under the operations of addition and scalar multiplication which were defined for V.
Your second question (If V is a linear space is it also a subspace?) is a bit difficult to answer because you did not specify what V is a subspace of? Trivially, V is a subspace of itself, but I don't think this is what you meant.
Let's consider a geometric example. Consider the vector spaces $\mathbb{R}$ and $\mathbb{R}^{2}$ defined in the usual way. Intuitively, many people want to say $\mathbb{R}$ is a subspace of  $\mathbb{R}^{2}$ but this is technically incorrect. we say instead that $\mathbb{R}$ is isomorphic to any 1-dimensional subspace of $\mathbb{R}^{2}$.
I hope this helps clarify the two concepts. As to why we bother with the concept of 2 different definitions, we can consider a set of homogeneous linear equations (things like 2x+3y+4z=0, the 0 on the RHS is what makes them homogeneous) in, say $\mathbb{R}^3$ and note that their solution set would form a subspace of $\mathbb{R}^3$
