Comparison between Van Kampen theorem from Allen Hatcher and "Introduction to knot theory".

The first big difference is that Hatcher's very general statement of Van Kampen's Theorem allows for an arbitrary number of members in the collection of subsets $\{A_\alpha\}$, whereas the special version in Crowell's book allows only for 2 members. The general proof requires direct limit arguments not present in the special version. So, let's specialize Hatcher's statement to the case of just 2 members:

If $X$ is the union of two path-connected open sets $A_1,A_2$ each containing the basepoint $x_0 \in X$ and if the intersection $A_1 \cap A_2$ is path connected, then the homomorphism $\Phi : \pi_1(A_1) * \pi_1(A_2) \to \pi_1 (X)$ is surjective. The kernel of $\Phi$ is the normal subgroup $N$ generated by all elements of the form $i_{12}(\omega) i_{21}(\omega)^{-1}$ for $\omega \in \pi_1(A_1\cap A_2)$, and hence $\Phi$ induces an isomorphism $\pi_1(X) \approx \pi_1(A_1) * \pi_1(A_2) / N$.

Now let's pick the two theorems apart. Hatcher's version says:

... the homomorphism $\Phi : \pi_1(A_1) * \pi_1(A_2) \to \pi_1 X$ is surjective...

whereas Crowell's version says

... the image groups $\omega_i G_i$... generate $G$...

and these are exactly equivalent statements if one takes $G_1 = \pi_1(A_1)$ and $G_2 = \pi_1(A_2)$.

Hatcher's version then described a specific normal subgroup $N < \pi_1(A_1) * \pi_1(A_2)$ and says that $N$ is the kernel of $\Phi$; whereas Crowell's version describes a diagram of groups, involving the group $G_0$ in particular, and says that a certain diagram involving the $G_i$'s satisfies a universal property. But, both are describing the same thing, namely the amalagamated free product. In translating between the two settings one would take $G_1 = \pi_1(A_1)$ and $G_2 = \pi_1(A_2)$, and one would take $N$ to be the subgroup of the free product $G_1 * G_2$ generated by elements of the form $\theta_1(z_k) \theta_2^{-1}(z_k)$ for $z_k \in G_0$ (Crowell writes $\bar\theta$ instead of $\theta$ for reasons that are unclear from the text you quote). The only difference here is between describing amalgamated free products via a universal property or describing them explicitly in terms of words and normal subgroups and quotient groups, which is what one learns in group theory.

So yes, once Hatcher's version of the theorem has been specialized to just two elements in the collection, the two theorems are pretty much identical. The translation of one to the other just requires a bit of group theory and category theory.

The reason seems to be that the books were written at different times and with different aims. One is intended, as you see from its title, as a book on knot theory, the other is a book on algebraic topology, more generally. Crowell was the first to publish in 1958 a comprehensible proof of a more general theorem, and gives a proof by direct verification of the universal property. The Preface of a $1967$ book by W.S. Massey stresses the importance of this idea. Van Kampen's 1933 paper is difficult to follow.

This universal property is not stated in Hatcher's version. The most general result of this type is in this paper which uses the fundamental groupoid on a set of base points, and so allows the computation of the fundamental group of the circle, and of a myriad of other spaces; also the proof generalises to higher dimensions, computing some relative homotopy groups: see this book.

Hope that helps.