Let me begin with the grand daddy of all invaraintsinvariants, the Euler characteristic of a triangulated space. This is an easily defined and computable invariant, but the price we pay for this ease in computation is that it is a rather rough tool. It cannot distinguish too many spaces. E.g., compact oriented $3$-manifolds have Euler characteristic $0$.
The construction of homology by Poincare can be viewed as a first instance of categorification. It takes a bit more work to define this new invariant but it is more powerful. The Euler characteristic of a space is the Euler characteristic of its homology. One advantage is obvious: the Euler characteristic could be trivial without the homology being so. Put it differently, the "sum" of all parts could be zero, but the parts themselves may not be so.
The Alexander and Jones polynomials are themselves Euler characteristics of more refined objects (Ozsvath-Szabo and resp. Khovanov homology). These polynomials could be trivial, but the corresponding homologies may not be so.
