Why do we need a Jordan normal form?

First of all, the Jordan form is also "good for computations and simplifying powers of matrices". In particular, if $$ J = \pmatrix{\lambda&1\\&\lambda&1\\&&\ddots & \ddots \\&&&&1\\&&&&\lambda} $$ Then we compute (when $k>n$) $$ J^k = \pmatrix{\lambda^k & k \lambda^{k-1} & \frac 1{2!} k(k-1)\lambda^{k-2} & \cdots & \frac 1{(n-1)!} k(k-1) \cdots(k-n + 1)\lambda^{k-n + 1}\\ 0 & \ddots & \ddots & \cdots\\ \vdots & \ddots} $$ More generally, if $f(x)$ is an analytic function (such as $f(x) = e^x$), we have

$$ f(J) = \pmatrix{f(\lambda) & f'(\lambda) & \frac 1{2!} f''(\lambda) & \cdots & \frac 1{(n-1)!} f^{(n-1)}(\lambda)\\ 0 & \ddots & \ddots & \cdots\\ \vdots & \ddots} $$ Jordan form is also important for determining whether two matrices are similar. In particular, we can say that two matrices will be similar if they "have the same Jordan form".

Exercise: using Jordan canonical form, prove that a matrix is diagonalizable (over C) iff the minimal polynomial has no repeated roots. Prove the Cayley Hamilton theorem (assuming Jordan form). (The idea in all cases is to understand these theorems for a Jordan block.) Prove that the determinant is the product of the eigenvalues, and that the trace is the sum.

So, you will see that the Jordan canonical form summarizes a lot of theorems as trivial corollaries. The ones I listed are just what came to mind, I'm sure there are others.

By the way - you will also notice that some of these theorems are trivial for diagonal matrices. There is a technique for proving theorems for diagonal matrices, and extending them to all matrices, since diagonalizable matrices are dense in the space of matrices, and so equalities of continuous functions will carry over to the whole space (you can even prove theorems in abstract linear algebra over any ring statement this way). All matrices have a JCF, so I guess the situations when you can prove stuff by reduction to JCF are strictly larger than when you can prove by reduction to diagonalizable matrix. (For example, the first exercise I listed.)

Generally, if we can find a canonical form for some object, it simplifies reasoning about the object. If we want to prove some fact, we need only deal with the canonical form rather than all possibilities.

The Jordan normal form is not quite canonical (the blocks can be permuted in general) but serves the same purpose. It shows how repeated eigenvalues can affect the eigenvalue structure.

Omnomnomnom's answer shows how analytic functions can be defined, other answers show how $\det$ and $\operatorname{tr}$ are related to the eigenvalues.

A word of warning, in absence of special structure (or exact arithmetic), it is impossible to numerically determine the Jordan form of a matrix if there are repeated or nearby eigenvalues.

Other forms are better for numerical work, for example, the Schur decomposition allows us to write a matrix in the form $QTQ^*$, where $Q$ is unitary and $T$ upper triangular. This form does not reveal the eigenspace structure as clearly as the Jordan normal form, but can be computed in a numerically stable fashion.