From what i have seen different places on the web $\nabla$ and $\vec \nabla$ is being used on many of the same things. Is there a difference?

I thought that $\nabla f$ was an operator on a function. And $ \vec\nabla f = \begin{pmatrix}\frac{\partial }{\partial x}\\ \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z}\end{pmatrix} f$ was a vector on a function / scalar.
In this example i would think i should get the same result. But if i look at
$\vec u = (u,v,w)= \begin{pmatrix}u\\ v\\ w\end{pmatrix}$ i get:

$\nabla \vec u = \begin{pmatrix}\frac{\partial u}{\partial x}\\ \frac{\partial v}{\partial y}\\ \frac{\partial w}{\partial z}\end{pmatrix}$ (My teacher said so). But i dont understand why this would not be $\nabla \vec u = \begin{pmatrix}\frac{\partial \vec u}{\partial x}\\ \frac{\partial \vec u}{\partial y}\\ \frac{\partial \vec u}{\partial z}\end{pmatrix}$. Wich is still a vector.

And if i take
$\vec\nabla \vec u = \begin{pmatrix}\frac{\partial }{\partial x}\\ \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z}\end{pmatrix} \begin{pmatrix}u\\ v\\ w\end{pmatrix}$ i get $ \begin{pmatrix}\frac{\partial }{\partial x}\\ \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z}\end{pmatrix} \begin{pmatrix}u\\ v\\ w\end{pmatrix} = \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}$ Wich is not a vector. And this seems to be $\nabla \cdot \vec u$

So from what i see and think should $\nabla \vec u = \begin{pmatrix}\frac{\partial \vec u}{\partial x}\\ \frac{\partial \vec u}{\partial y}\\ \frac{\partial \vec u}{\partial z}\end{pmatrix} = \begin{pmatrix}\frac{\partial u}{\partial x}&\frac{\partial v}{\partial x}&\frac{\partial w}{\partial x}\\ \frac{\partial u}{\partial y}&\frac{\partial v}{\partial y}&\frac{\partial w}{\partial y}\\ \frac{\partial u}{\partial z}&\frac{\partial v}{\partial z}&\frac{\partial w}{\partial z}\end{pmatrix} = \vec \nabla \vec u^{T} = \begin{pmatrix}\frac{\partial }{\partial x}\\ \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z}\end{pmatrix}\begin{pmatrix}u&v&w\end{pmatrix}$. What is wrong with this or why is it correct?