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!

Yes, a subspace is a space in itself. The conditions required for a subspace $+$ the conditions that the subspace inherits from the space make it a space in it's own right. 
You should check here.
Why are they important? A subspace lives inside a space. The fact that it has in itself all the nice properties of a space make it interesting. For example, if you have a space $V$ and your subspace $W$ is generated by linearly independent vectors $v_1, v_2, ... , v_k$, then any vector that is a linear combination of these $k$ vectors also belongs to the same subspace. 
(Sorry about those comments, pressed enter by accident)
