Topology studies spaces up to homeomorphism. Algebraic topology invents coarser invariants, like homotopy type and homology group. These will help distinguish spaces that are sufficiently different, and allows the use of algebraic machinery. Distinguishing spaces is the same as showing that there is no homeomorphism between them. More generally, these algebraic techniques enable one to show that maps with certain properties do not exist (because they would lead to morphisms of homotopy or (co)homology groups, and by algebra these do not exist). A famous example is the Brouwer fixed point theorem:
Theorem Let E(n) be the n-ball in Euclidean n-space. Then every continuous map from E(n) into itself has a fixed point.
This is an immediate corollary of
Theorem Let S(n-1) be the (n-1)-sphere in Euclidean n-space, the boundary of E(n). If n > 0 then the identity map from S(n-1) to itself has no extension to a map from E(n) to S(n-1).