I'm looking for manners in which the 2-sphere (the regular 3-dimensional object we encounter in real life) is qualitatively different from the n-spheres. Of course it is the only one among those that is a 2-dimensional manifold, but I'm looking for something less trivial.
As an example, a qualitative difference between the 1-sphere (a circle) and the 2-sphere is that the 1-sphere is not simply connected.
So the question would be: Is there some property X - that doesn't in some round about way talk about dimension - such that among the n-spheres, the only one that has property X is the 2-sphere.
A related question would be if there is some qualitative difference between SO(3) and SO(n).
Here's an answer, but I don't know whether it will make you happy or not. Since you gave "simply connected" as an example of the kind of property you're interested in, there's an obvious generalization to higher dimensions: A space is said to be $n$-connected if every continuous map from $S^k$ into the space can be continuously shrunk to a point whenever $0\le k\le n$. Thus $1$-connected is the same as simply connected. Among spheres, $S^n$ is the unique one that's $(n-1)$-connected but not $n$-connected.