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).