Consider the complex matrices $$ \begin{array}{c} A=\left(\begin{array}{ccc} -2 & 4 & 3 \\ 0 & 10 & 9 \\ 0 & -16 & -14 \end{array}\right) \qquad \text{ and }\qquad B=\left(\begin{array}{ccc} -4 & -5 & -2 \\ 0 & -2 & 0 \\ 2 & 5 & 0 \end{array}\right) \end{array} .$$ Determine whether $A$ is similar to $B$.
Attempt: I am tempted to say they are similar since they have the same rank, characteristic polynomial (and therefore determinant, trace and eigenvalues with algebraic multiplicities), geometric multiplicities of eigenvalues, minimal polynomial and Jordan normal forms, up to a permutation of the Jordan blocks, but on the other hand I think they are necessary but not sufficient conditions for similarity.
You state that the two matrices have the same Jordan normal form. But every matrix is similar to its Jordan normal form.