Is $\mathbb{R}^n$ properly homotopy equivalent to $\mathbb{R}^m$ if $n \neq m$?

The invariants which distinguish the different $\mathbb{R}^n$'s up to proper homotopy are the "homotopy groups at infinity". For example, let's distinguish $\mathbb{R}^2$ from $\mathbb{R}^n$ with $n \ge 3$ by using "simple connectivity at infinity".

For every sequence of nonempty compact sets $K_1 \subset K_2 \subset \cdots \subset \mathbb{R}^2$ whose union equals $\mathbb{R}^2$, there is exactly one component $U_i$ of $\mathbb{R}^2 - K_i$ whose closure is noncompact, we have an inclusion $U_1 \supset U_2 \supset \cdots$, and we have a sequence of injections of fundamental groups of the form $$\pi_1(U_1) \leftarrow \pi_1(U_2) \leftarrow \cdots $$ The inverse limit of this sequence of groups is well-defined up to isoomorphism, and it is isomorphic to $\mathbb{Z}$.

But if you do the same thing with $\mathbb{R}^n$ where $n \ge 2$, the inverse limit of the sequence of groups will be trivial.

Of course one still has to prove that this inverse limit is a proper homotopy invariant, and one has to prove this for higher homotopy groups, and one has to do the appropriate calculations for the $\mathbb{R}^n$. The upshot is that the first nontrivial homotopy group at infinity for $\mathbb{R}^n$ is the $n-1^{\text{st}}$.

Here is a paper of Davis and Meier where you'll find some details about homotopy groups at infinity.

In algebraic topology we want to find invariants to distinguish spaces up to certain equivalences. For proper homotopy equivalence one of such is compactly supported cohomology. By Poincaré duality for $0\neq n\neq m$:

$$ H_0(\mathbb R^n) \cong H^n_c(\mathbb R^n) \not \cong H^n_c(\mathbb R^m) =0 $$