In one place, the theory of fibrations, I feel tom Dieck's book is superior. Thus in the first edition of Spanier, in the proof of Lemma 11, of Chapter 2, Section 7, Spanier writes down an extended lifting function $\Lambda$, but he does not prove it is continuous. I did not manage to prove it was continuous, and in fact found the function was not well defined. I sent a correction of the definition to Spanier, and this appeared in the second edition, but I still do not know how to prove $\Lambda$ is continuous. Have I missed something?
On the other hand Spanier's ideas on the construction of covering spaces are still referred today by experts, with the notion of what they call the Spanier group.
I find Section 3 of Chapter 7 of Spanier on "Change of base point" rather hard work, and I feel it can all be much easier done, and more generally, by using fibrations of groupoids. But tom Dieck's book does not use this method either.
Spanier gives van Kampen's theorem for the fundamental group of a simplicial complex as an exercise, while tom Dieck's book does give the statement of the theorem for a union of two spaces. Hatcher gives a more general theorem, for a union of many spaces, but none of these mention the fundamental groupoid on a set of base points.
I tend to agree with 313's answer that readers should look around, and find what is easier for them, in different aspects.
My copy of Spanier, dated 1966, had a price of \$15, but when I checked for inflation that was equivalent to \$111 today.
March 13: I add that neither book develops the algebraic theory of groupoids. For a relevant discussion on this, see https://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one/46808#46808
Let me convert my comment to a full answer:
Unless one is (and you are not!) planning to write a PhD thesis in General Topology, Munkres is (more than) enough.
Depending on what you are planning to study later, you might encounter an issue requiring a bit more General Topology (e.g. proper maps and proper group actions, which you will find in Bourbaki), but you learn this on "need to know" basis (just pick up a General Topology book and look it up when necessary). Instead, my suggestion is to start reading Guillemin and Pollack, and Hatcher (or Massey).
In addition, you would want to (or, rather, have to) learn more functional analysis (say, Stein and Shakarchi) and PDEs (say, Evans) which will be handy if you are planning to go into modern differential topology (which most likely will require you dealing with nonlinear PDEs, believe it or not), and, in case of algebraic topology, - basic category theory (at least be comfortable with the language), Lie theory (at least to know the basic correspondence between Lie groups and Lie algebras), see suggestions here. Yes, General Topology is fun and there are many neat old theorems that you will learn by studying it in more detail, but you have to prioratize: Life is short and your time in graduate school is even shorter.
Best Answer
I'm with Jonathan in that Hatcher's book is also one of my least favorite texts. I prefer Bredon's "Topology and Geometry."
For all the people raving about Hatcher, here are some my dislikes: