Never say never. There are cases where you want to connect with the ability to run arbitrary queries, for example to update a db schema at run-time or the like.
I think it would be more correct to say that applications should connect with the least privilege they need.
The first step is always to create the database, unless you want to use one from a third party. When a database is created, it is assigned to an owner, who executed the creation statement. Usually, only the owner (or a superuser) can do anything with the objects in that database, and in order to allow other users to use it, privileges must be granted.
Applications should never connect to the database as its owner or a superuser, because these users can execute any query at will, for example, modifying the schema (e.g. dropping tables) or deleting its entire content.
You may create different database users for every aspect of your application with very limited rights to database objects. The most required privileges should be granted only, and avoid that the same user can interact with the database in different use cases. This means that if intruders gain access to your database using your applications credentials, they can only effect as many changes as your application can.
You are encouraged not to implement all the business logic in the web application (i.e. your script), instead do it in the database schema using views, triggers or rules. If the system evolves, new ports will be intended to open to the database, and you have to re-implement the logic in each separate database client. Over and above, triggers can be used to transparently and automatically handle fields, which often provides insight when debugging problems with your application or tracing back transactions.