There are geometric and topological properties that significantly distinguish between Cartesian spaces of odd and even dimensions. Take, for example, the "sphere combing" property. The $(n-1)$-dimensional sphere (the boundary of a ball) in $\mathbb{R}^n$ admits a non-vanishing continuous tangent vector field if and only if $n$ is even.
Another striking difference is related to the existence of Hadamard matrices. It is known that an $n\times n$ Hadamard matrix exists only if $n=1,\ 2$, or is a multiple of $4$ (see https://en.wikipedia.org/wiki/Hadamard_matrix ), and Hadamard conjectured that there exists such a matrix for every $n$ divisible by $4$. The conjecture is still unsolved, but there exist many constructions producing an $n\times n$ Hadamard matrix for infinitely many values of $n$. Now, the geometric connection is this: Some $n+1$ vertices of the $n$-dimensional cube form a regular simplex if anand only if there exists an $(n+1)\times(n+1)$ Hadamard matrix.
Thus, spaces $\mathbb{R}^n$ and $\mathbb{R}^{n+1}$ often differ essentially for infinitely many values of $n$, and I think many other examples of this nature can be readily presented.