Translation matrix without homogeneous coordinates I am looking to translate a vector, and I found this article on wikipedia, which states that I need to use homogeneous coordinates to do so.
I don't get the reason why, and I don't really see how the result of the translation matrix with homogeneous coordinates is different from a translation matrix
$$ \matrix{T} =
\begin{bmatrix}
v_x & 0 & 0 \\ 
0 & v_y & 0 \\
0 & 0 & v_z \\
\end{bmatrix}
$$
so that with my vector $\vec{p} = [p_x \, p_y \, p_z]^T$ I'd get the translation as
$$
\vec{r}=\matrix{T}\vec{p}=
\begin{bmatrix}
v_x+p_x & 0 & 0 \\ 
0 & v_y+p_y & 0 \\
0 & 0 & v_z+p_z \\
\end{bmatrix}
$$
What is the reason for using the homogeneous coordinates instead of the above?

As you say, you don't really need to use homogeneous coordinates to represent translations. You can translate a point $\mathbf{p}$ by a vector $\mathbf{v}$ just by adding coordinates: $T(\mathbf{p}) = \mathbf{p} + \mathbf{v} = (p_x+v_x, p_y+v_y, p_z+v_z)$.
The benefit of using homogeneous coordinates and $4 \times 4$ matrices is that they give you a way to represent many different kinds of geometric transforms in a uniform way. Specifically, translations, rotations, scaling, shearing, and even central (perspective) projections can all be represented this way. 
There are a couple of reasons this is valuable:
Firstly, if you need to compose transformations (perform several in succession, one after another), you can do this just by multiplying matrices. No special-case thinking is required. Life is tidier.
Secondly, if you're building graphics hardware, then all you need to do is create devices that can multiply $4 \times 4$ matrices, and these devices will then be able to perform a wide variety of different transformations, including all the ones that are needed in computer graphics. Hardware devices like uniformity.
