There is a well-developed technology for computing the cohomology groups of a CW complex, cellular cohomology. It reduces the problem of computing cohomology to the two simpler problems of (1) endowing a space with a CW structure and (2) computing the degrees of some maps between spheres. Consequently, the problem of computing the cohomology of a CW complex is completely solved and effectively computable.
To contrast, the homotopy groups of a CW complex are not effectively computable. Famously, even the simple case of the homotopy groups of the spheres is extremely difficult and not completely resolved.
What about the cup product structure on cohomology? Is the cohomology ring effectively computable? Is there any "algorithm" to compute the cohomology ring of a CW complex, similar to the "algorithm" of cellular cohomology?
I would guess that the answer to these questions is "yes" because cohomology seems eminently computable. Is it?