The best elementary justification for algebraic topology is that it provides a way of rigorously proving when some spaces are not homeomorphic. It's intuitively obvious, for instance, that the 2-dimensional sphere $S^2$ and the torus $T^2$ are very much different spaces. But to actually prove from scratch that every map $S^2\to T^2$ is either not bijective or has a discontinuous inverse is daunting. On the other hand, as soon as you know the first facts of homology theory or about the fundamental group you can write down a proof in just a couple of lines.
A somewhat more sophisticated motivation is that algebraic topology has the best tools for making sense of questions that are invariant up to homotopy-most simply, which spaces are homotopy equivalent. The fundamental group (and higher homotopy groups) and (co)homology theory (theories) are just as good at distinguishing non-homotopy equivalent spaces as non-homeomorphic ones, which at first sight is even harder to do. Now, you'll notice I haven't said anything about proving spaces are homeomorphic, or homotopy equivalent. The former is impossible in generality much beyond, say, 2-dimensional manifolds. But the latter is possible, at least in theory, for a kind of space called CW-complexes, which are thus a favorite of algebraic topologists.
The point, overall, is that algebraic topology provides one with discrete invariants that are more tangible material for writing rigorous proofs than the purely topological motivation for an idea-it's much harder to fully comprehend a space, especially in words, than a group associated to that space. This reflects the common pattern that algebra is more verbal and geometry more visual.
I can't comment very well on the dividing line between general topology and algebraic topology, because I don't know anything at all about modern topological research that's not either algebraic or geometric (i.e. about manifolds.)