What is the purpose of naming your constraints (unique, primary key, foreign key)?
Say I have a table which is using natural keys as a primary key:
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By naming the constraints you can differentiate violations of them. This is not only useful for admins and developers, but your program can also use the constraint names. This is much more robust than trying to parse the error message. By using constraint names your program can react differently depending on which constraint was violated.
Constraint names are also very useful to display appropriate error messages in the user’s language mentioning which field caused a constraint violation instead of just forwarding a cryptic error message from the database server to the user.
See my answer on how to do this with PostgreSQL and Java.