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?

As others have noted, they are identical. If you would like to see the equality in practice, consider the following:
Theorem: If $A$ is an $m\times n$ matrix with real coefficients, then there exists an $n\times m$ matrix $B$ such that 
$$(Ax)\cdot y = x\cdot(By)$$
for all $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^m$.
Proof: By your equality:
$$(Ax)\cdot y = (Ax)^Ty = x^TA^Ty = x\cdot(A^Ty).$$
So, $B=A^T$.
