It is generally difficult to show that two spaces are homotopy equivalent. Some of the better spaces are CW complexes, where the Whitehead theorem holds.
In this case, you only need that there be a map that induces an isomorphism on all of the homotopy groups in order to have a homotopy equivalence. As was mentioned in the comments, you still need a map that realizes this. It is not often the case that just knowing the isomorphism type of some (easily-computable) invariant gives you the homotopy type of a space.
Sometimes we luck out and we have very good invariants, e.g.
$$\{ \chi \text{ and orientability} \} \leftrightarrow \{ \text{homotopy types of closed surfaces} \}$$
but this is very rare.
A nice situation that is maybe worth mentioning is for rational spaces. Such spaces have a "super Whitehead theorem":
Theorem. Suppose that $X, Y$ are nilpotent spaces with homotopy groups that are finite dimensional rational vector spaces. Then the following are equivalent for a map $f: X \rightarrow Y$:
- $f$ is a homotopy equivalence;
- $f_*: H_* (X, \mathbb{Q}) \rightarrow H_*(Y, \mathbb{Q})$ is an isomorphism;
- $f_*: \pi_* (X) \rightarrow \pi_* (Y)$ is an isomorphism.
So, we see that in fact all one needs is a homology isomorphism in order to detect that a map is a homotopy equivalence, provided that the spaces are rational. Of course, you still need that this isomorphism is realized by a genuine map, as abstract isomorphism will not suffice to have homotopy equivalence.
Hint: for $n\geq 2$, the $n$-sphere $S^n$ is simply connected, and thus, any map $f:S^n\to T^n$ admits a lift to the universal cover of the torus, that is to say, there is a map $F:S^n\to \Bbb R^n$ such that $f=\pi\circ F$ where $\pi:\Bbb R^n\to T^n$ is the universal cover of the torus.
Best Answer
In general it is extremely hard to calculate the set of homotopy classes of maps between two spaces, and the problem does not reduce to understanding their homotopy groups. This problem has as special cases questions like 1) how to compute an arbitrary cohomology group of an arbitrary space, 2) how to compute an arbitrary homotopy group of an arbitrary space, 3) how to classify vector bundles on an arbitrary space, etc.
For the very special case of maps from a closed orientable $n$-manifold (such as the torus $T^n$) to a sphere $S^n$ of the same dimension we are very lucky: by the Hopf theorem such maps are classified by their degree, so there are a $\mathbb{Z}$'s worth of them.