In linear algebra I've seen that there are two major different ways to write the characteristic polynomial of a mapping $f$: As $$ p_{f}=\pm\left(T-\lambda_{1}\right)^{m_{1}}\cdot\ldots\cdot\left(T-\lambda_{k}\right)^{m_{k}} $$ with distinct $\lambda_{i}$ or as $$ p_{f}=\pm\left(T-\lambda_{1}\right)\cdot\ldots\cdot\left(T-\lambda_{n}\right) $$ where the $\lambda_{i}$ are not necessarily distinct.
The only advantage I can see the latter expression has over the former is that the eigenvalues are already given in an order from $1$ to $n$, so it's easier to user the latter expression when doing something like induction over the eigenvalues.
But this argument is not very convincing to me...what kind of situations (theorems) do you know, where it is definitely better to use one expression over the other ? What are the disadvantages of each of the expressions ?
In bringing the matrix to Jordan canonical form (see http://en.wikipedia.org/wiki/Jordan_canonical_form) it is crucial to have the multiplicities you denoted $m_i$ so as to describe the blocks into which the matrix decomposes (up to conjugation).
On the other hand, computationally speaking the coincidence of eigenvalues is not stable, and from this point of view the significance of the Jordan canonical form is purely theoretical. From the viewpoint of applications the second form may be more relevant.