Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results?
If they are identical, then I suppose the only difference between them is the method of calculation, eh?
See Wikipedia for all definitions. Take this matrix: $$ \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix} $$ Its Frobenius norm is $\sqrt{10}$, but its eigenvalues are $3,1$ so, if the matrix is symmetric, its $2$-norm is the spectral radius, i.e., $3$. The Frobenius norm is always at least as large as the spectral radius. The Frobenius norm is at most $\sqrt{r}$ as much as the spectral radius, and this is probably tight (see the section on equivalence of norms in Wikipedia).
Note that the Schatten $2$-norm is equal to the Frobenius norm.