It seems like any countable, non-dense subset of the reals is a vector by any other name. Don't get me wrong, a set is a set, and a vector is a vector - it just seems like there isn't much to distinguish the two aside from name and context.
Consider, for example, the set of natural numbers $\mathbb{N}=\{1,2,\ldots\}$. In what way is the vector $\textbf{v}\in\left\{\mathbb{N}^\infty\mid v_n=n\right\}, \textbf{v}=(1,2,\ldots)$ not the same as $\mathbb{N}$? Both have the same number of elements, satisfy total order, contain the natural numbers, and it is possible to substitute one for the other in a great many cases with little to no effort (ex: $\sum_{n\in\mathbb{N}}f(v_n)=\sum_{n\in\mathbb{N}}f(n)$, $a_n=g(v_n)=g(n)=\left(a_n\right)_{n\in\mathbb{N}}$, etc.)
I suppose that the representation of a vector, in writing, is understood to imply its order (whereas the order of the terms in a set do not change the intended meaning) - but I feel like there should be something more substantial than this. After all, the written representation of a mathematical object is distinct from the object itself, and I could easily create a notation where the written order of terms in a vector need not correspond to that of the actual vector.
I hope this isn't a silly question.
To clarify: This is not to say that a set and a vector are the same thing. Rather, the information needed to describe a vector is the same information needed to describe a particular type of set (i.e. the ordering between any two members)
Unfortunately, this probably is not a satisfying answer, but the key difference between the set and the vector is that the set has no order to its terms, while the vector does. You say you could change the written representation of the vector and change the order, but that change in the written representation does not alter the fact that the inherent mathematical object which that representation represents—that is, the vector—still has an order in its underlying structure.
I think a better way to explain this is by trying to better understand what $\Bbb{N}^\infty$ is. For me, I like to think of $\Bbb{N}^\infty$ as the set of all functions from $\Bbb{N}\rightarrow \Bbb{N}$. Thus, here, $v$ is the vector such that $v(n)=n$ for all $n\in \Bbb{N}$, which is like the identity function on $\Bbb{N}$. However, even though $v(n)$ can always just be replaced with $n$ and therefore the identity function may seem redundant, that does not change the underlying structure behind $v(n)$, which is that it is an infinite-dimensional vector, and not a set.
Another way to realize the difference is by examining the vector $w(1)=2$, $w(2)=1$ and $w(n)=n$ for all $n > 3$. Clearly, the set of components of this vector are the natural numbers. However, this is also true for $v$, yet $v\neq w$ because $v$ and $w$ have their components ordered differently: $v$ has the order $1$, then $2$, while $w$ has the order $2$, then $1$. On the other hand, $\Bbb{N}$ has no such underlying order to it, so it is fundamentally distinct from vectors.