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[[ Sorry I missed that the question was also concerned with the question in an algebraic topological context. This answer is only concerned with algebraic geometry.]]

I think the first question is much easier to answer. mdeland has given the Artin-Mumford non-rationality example as one answer. Another is the Atiyah-Hirzebruch example of an even-degree torsion class of a smooth projective variety which is not algebraic, showing that an integral version of the Hodge conjecture is false. This gives examples (and there are others) where torsion can be used to show something about an algebraic variety which one couldn't show without (actually I would say that it is more a question of integral versus rational cohomology even without torsion one can exploit that certain cohomology classes are not divisible by some particular integer). I would say that gives an answer to the first question.

The second is of a very different nature. In algebraic topology torsion (and more general integral cohomology again versus rational cohomology) are enormously important for understanding the homotopy type of a space. Take as an example the spheres. Rationally their homotopy theory is trivial but integrally you have highly non-trivial homotopy groups (this non-triviality does not reflect itself in the cohomology of the spheres but is closely related to spaces derived from the spheres, the pieces of the Postnikov tower). Of course algebraic varieties (over $\mathbb C$, but that is not essential) give homotopy types too but it not always clear what the homotopy type of an algebraic variety tells you about the algebro-geometric structure of the variety (unless you somehow incorporate algebraic topology under algebraic geometry...). There are some examples though: The torsion in the second cohomology group comes directly from the fundamental group and in particular give you abelian étale covers of the variety. The torsion in the third cohomology group tells you about the Brauer group of the variety and in particular corresponds (for some definition of "corresponds") to projective fibrations over the variety. The correspondence is quite indirect however. I would for instance love to know the least relative dimension of a projective fibration over an Enriques surface which realises the element of order $2$ in the third cohomology group or even better a geometric construction of any such fibration. In higher cohomological degrees the situation is even worse (unless one chooses the above incorporation option, higher algebraic stacks could be said to do that).
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