Corruption is when something is actually wrong with the database. For example a page is corrupt and cannot be read anymore from disk.

Inconsisteny is a problem within the data itself and the way it has been modelled. But the database itself can still be perfectly fine.

For example, two tables store customer information. Dave has been stored twice with the same email address. One table is updated and the email address gets a new value. The database is now inconsistent, because Dave now has two possible email addresses, but which one is correct? The risk for inconsistency can be minimized by correctly normalizing the model. Another important factor for inconsistency are transactions. If a transactions fails halfway through and doesn't roll back, it can leave the database in an inconsistent state. That's why transactions need to adhere to the ACID model, where the C stands for consistency.