What is the difference between $n$-tuples, $m \times 1$ and $1 \times n$ matrices? Isn't the tuple different structure from $m \times 1$ or $1 \times n$ matrix? Why can you mix them? 
 A: From Wikipedia, "A tuple is a finite ordered list of elements." So things like $(\clubsuit, \square, 57, \clubsuit)$ are perfectly valid tuples, and we can view this particular tuple as one function
$$f: [4] = \{1, 2, 3,4\} \to A,$$
where $A$ is just some set that includes $\{\square, \clubsuit, 57\}$ as a subset. We relate this tuple to $f$ by considering it as a list of outputs of our function: $\big(f(1), f(2), f(3), f(4)\big) = (\clubsuit, \square, 57, \clubsuit)$. So tuples are really the most general.
Now, when we consider only functions
$$f: [m] = \{1, 2, \ldots, m\} \to \Bbb F$$
that map into some field $\Bbb F$, we get very special tuples (remember: a field is a set of "numbers" in which we can add and multiply, and where all non-zero "numbers" have a multiplicative inverse, like the real numbers $\Bbb R$ or the complex numbers $\Bbb C$).
If we write such tuples as 
$$v = \begin{bmatrix}f(1) \\ f(2) \\ \vdots \\ f(m)\end{bmatrix},$$
we have an $m \times 1$ matrix, or column vector, of the vector space $\Bbb F^m$.
If, on the other hand, we write such tuples as 
$$v = \begin{bmatrix}f(1) & f(2) & \cdots & f(m)\end{bmatrix},$$
we have an $m \times 1$ matrix, or row vector, of the vector space $ (\Bbb F^m)^*$. 
If we're being pedantic, one difference between a generic tuple and row/column vectors is that vectors always belong to an algebraic structure (a vector space), while this may or may not be true for an arbitrary tuple.
We can even distinguish between row and column vectors, algebraically. For example, given 
$$v = [1, 2] \in \Bbb R^2, \quad v^t = \begin{bmatrix}1 \\ 2\end{bmatrix} \in (\Bbb R^2)^*,$$
then given a $2 \times 2$ matrix such as $M = \begin{bmatrix}1 & 3 \\ -2 & 5\end{bmatrix},$ then $Mv$ makes sense while $vM$ doesn't, and $v^tM$ makes sense, while $Mv^t$ doesn't (and further, $Mv$ is probably not equal to $v^tM$. To learn more, read about Linear Algebra and vector spaces).
So, "as a list of $m$ things", tuples, row vectors, and column vectors are all "the same", we get bijections between all three (and structure-preserving maps between a vector space $\Bbb F^m$ and its dual $(\Bbb F^m)^*$). And while each fills a separate mathematical role, generally we may gloss over the details the details and the line gets blurry.
A: There is a difference because of the definition of those objects (I take $\mathbb R$ as the underlying number system):


*

*A $m\times 1$ matrix represents a linear function $\mathbb R^m \rightarrow \mathbb R$.

*A $1\times m$ matrix represents a linear function $\mathbb R \rightarrow \mathbb R^m$.

*A $m$-tuple represents a finite sequence of real numbers of the length $m$.


But they have a key property in common:

A $m\times 1$ matrix, a $1\times m$ matrix and a m-tuple can be uniquely defined by giving $m$ real numbers.

Or better to say:

The set of all $m\times 1$ matrices, the set of all $1\times m$ matrices and the set of all m-tuples are vector spaces of dimension $m$

Thus all vector spaces are isomorph (because there dimension is the same). Note that isomorph means "have the same structure". Thus you can identify a $m\times 1$ matrix with a $1\times m$ with a $m$-tuple.
