Are there reasons not to use product of vectors as dot product? The dot product of two vectors is defined as following:
$$
\langle \vec v, \vec u \rangle = 
\left<
\begin{pmatrix}
v_1 \\
v_2 \\
\dots \\
v_n
\end{pmatrix},
\begin{pmatrix}
u_1 \\
u_2 \\
\dots \\
u_n
\end{pmatrix}
\right>
= v_1 \cdot u_1 + v_2 \cdot u_2 + \dots + v_n \cdot u_n
$$
Still the multiplication of transposition of $\vec v$ and u gives:
$$
\vec v^T \cdot \vec u = (v_1, v_2, \dots, v_n) \cdot
\begin{pmatrix}
u_1 \\
u_2 \\
\dots \\
u_n
\end{pmatrix}
= v_1 \cdot u_1 + v_2 \cdot u_2 + \dots + v_n \cdot u_n
$$
so the result is the same!
It may be just a silly observation but I'm just surprised because I have never seen using it.
Are these two notations the same thing or is there something important in their definitions that don't allow interchanging them?

Strictly speaking, a row vector represents a linear form, i.e.  a lineap map from $\mathbf R^n\to\mathbf R$. So they're different in essence.
However, to the vector $\vec v$, you can associate the linear map 
\begin{align*}\varphi_{\vec v}\colon\mathbf R^n&\longrightarrow\mathbf R\\\vec u&\longmapsto \langle\vec v,\vec u\rangle \end{align*}
and in this association, the column vector that represents $\vec v$, becomes its transpose. So it is quite normal the results are the same.
