First of all there is no much difference, because every subspace IS a vector space. Also, every vector space IS a subspace (of itself).
The difference occurs only when we are asked to VERIFY in practice, that some set is a vector space. A SUBSPACE S can be defined as a subset (=a part of) a VECTOR SPACE V which is also a vector space, WITH THE SAME ADDITION AND MULTIPLICATION as in V.
In one type of problems, we want to verify that certain set with certain operations is a vector space. Then we have to verify that the operations have properties 1-8 of the definition of a vector space.
But in other problems a vector space V is already GIVEN, which means that the operations are already defined, and we KNOW that they have properties 1-8. We are asked to verify, that certain part S of V is also a vector space with these same operations. Because the operations are the same as in V, we don't need to verify properties 1-8 again, but we only need to verify that whenever x and y belong to S, also x+y and cx belong to S. (If this is the case, we also say that the operations from V are WELL DEFINED in S).
For example, in Probl. 8, p. 313 we are given some unusual operations, we've never seen before, so we have to verify that they have properties 1-8. If they do, K is a vector space, if they don't it is not.
Problem 10 on the same page is of second kind: we are given a part Q of the vector space of all functions, with usual operations. We already know (this was mentioned in class) that the set of all functions with usual operations is a vector space (this means that usual operations on functions have properties 1-8). So we don't have to verify this again, the only thing to verify is that these usual operations are well defined on Q, that is that a sum of 2 elements of Q is always an element of Q, and a product of an element of Q with a number is always an element of Q.
Problem 9 is like problem 8 (some new weird operations are given).
Problems 2,3,5,6,7,10,11,12 are of the second kind, because all those sets mentioned are subsets of the set of all functions, with usual operations.
P.S. I am always mentioning "properties 1-8", because there were 8 numbered properties in the definition given in class. The book has 10 of them, by adding 1 and 6. In my formulation 1 and 6 are contained in the words that "operations are defined". In fact it is 1 and 6 (and only these two) which you have to verify to check that something is a subspace of a given space.