For a vector $x \in \mathbb{R}^n$,
$$ \|x\|_p = \left( \sum_{i=1}^n |x_i|^p \right)^{1/p}$$
gives the standard Euclidean norm when $p=2$, i.e., $\|x\|_2 = \left(\sum\limits_{i=1}^n x_i^2\right)^{1/2}$.
For an $m \times n$ real matrix $A$, we define the Frobenius norm as follows.
$$\|A\|_F = \operatorname{Tr} \left(^t\!A\ A \right) = \operatorname{Tr} \left( A\ ^t\! A \right) = \Big(\sum_{i=1}^m \sum_{j=1}^n a_{i,j}^2\Big)^{1/2}$$
i.e., $\|A\|_F$ is nothing else than the $\|\cdot\|_2$ vector-norm of $A$ seen as a vector of $\mathbb{R}^{m\times n}$. But the $\|\cdot\|_2$ matrix-norm is different in general:
$$\|A\|_2 = \sup_{x \neq 0} \frac{\|A x\|_2}{\|x\|_2} \neq \|A\|_F$$
These notations can seem quite confusing, what is the motivation for not using the notation $\|\cdot\|_2$ for the Frobenius-norm which is just the $\|\cdot\|_2$ vector-norm of $A$ seen as a vector of $\mathbb{R}^{m\times n}$ ?
It is confusing, and one wants the simplest notations for the most useful concepts, and different people and different subjects find different concepts to be more useful.
But: For many, the operator (or matrix) norm $\|A\| = \sup_{x\ne0} \|Ax\|/\|x\|$ is much more important than the Frobenius norm. So it gets the simpler notation.