I updated beaTunes and now my key does not work anymore

When updating from beaTunes 1 to beaTunes 2 your license key for beaTunes 1 will stop working. The same is true for the step from 2 to 3 and from 3 to 4.
If you still want to update, you do not need to buy a completely new license. Instead there is usually a specially priced update license on the beaTunes purchase page.

beaTunes uses a major.minor.micro versioning scheme.
E.g. beaTunes 2.0.3 has a major version of 2, a minor version of 0 and a micro version of 3.

Micro and minor updates are always free.
Major updates are not, but there is usually a special update price available for those who want to buy a license 1 major version greater than the one they own.

For example:

An update from X.Y.Z to X.A.B is free (same major version)
An update from X.A.B to Y.A.B is not (different major version)