Suppose I have 
$$ A = 
\begin{bmatrix}
a\\b\\c\\d\\
\end{bmatrix}$$

$$ B = 
\begin{bmatrix}
a& b& c & d\\
\end{bmatrix}$$

Now, I know $A = B^T$. But in what sense are these different mathematical objects? 

$$
\begin{bmatrix}
1\\
2\\
3\\
4\\
\end{bmatrix} + \begin{bmatrix}
1\\
2\\
3\\
4\\
\end{bmatrix}= \begin{bmatrix}
2\\
4\\
6\\
8\\
\end{bmatrix}
$$

$$
\begin{bmatrix}
1&2&3&4\\
\end{bmatrix} + \begin{bmatrix}
1&2&3&4\\
\end{bmatrix}= \begin{bmatrix}
2&4&6&8\\
\end{bmatrix}
$$

To me, these seem to behave the same way. Is there any difference between a row vector and a column vector? How are $A$ and $B$ different?