When are $\mathbb{R}^n$ and $\mathbb{R}^m$ essentially similar? Here is a rather vague and subjective question: for which $n$ and $m$ are 
$\mathbb{R}^n$ and $\mathbb{R}^m$ ``essentially similar''?  The answer depends partly on what type of
mathematician is answering it.
Of course, in a certain technical sense, $\mathbb{R}^n$ and $\mathbb{R}^m$ are different for any $n\neq m$ since they 
are not homeomorphic (or linearly isomorphic); thus they by definition differ in some describable topological (or algebraic) way.
But qualitatively all Euclidean spaces, especially ones of large dimension, seem to be ``similar.'' 
The question is really: how large does $n$ have to be for Euclidean spaces of dimension $\geq n$
all to behave essentially the same way?
I have included my own discussion of some possible answers in my Real Analysis manuscript at http://wolfweb.unr.edu/homepage/bruceb/Meas.pdf (Section XI.18.3).  I invite comments on this discussion.
 A: 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 and 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.
A: The qualitative intuition that you mention — that $\mathbb{R}^n$ begin to be similar in sufficiently high dimension — is the same intuition underlying my question Does the truth of any statement of real matrix algebra stabilize in sufficiently high dimensions? from some time ago.
Specifically, I had asked whether every statement in the language of what I called real-matrix algebra eventually stabilizes in $\mathbb{R}^n$ for sufficiently high dimension. The language has sorts for scalars, matrices, row vectors and column vectors and operations for the natural additions and multiplications. The language does not allow one to make explicit reference to the dimension. I had thought that statements in this language might stabilize in high dimension. 
But the answers showed that that was wrong, and that one can find statements whose truth values do not stabilize in high dimension, but rather detect various number-theoretic features of $n$. 
Thus, those answers tend to refute the intuition that even for very simple statements, $\mathbb{R}^n$ begins to look alike in all sufficiently large dimension. 
A: I remember there were some speculations by Yurii Manin that our arithmetic is impect because it distinguishes between huge numbers that differ by 1 which should for all practical purposes be the same. I am not sure I follow Manin here but anyway he did make such an argument, though for the moment I can't find it in MathSciNet. Perhaps this did not reach MSN, and perhaps in the context of physics this actually makes sense. At any rate this seems like an intriguing question, though in my view the answer is negative: you can have an infinite prime number $H$ which is unlike $H+1$.
