They are not only different, they are orthogonal.
Inconsistency:
A DB that reported employee Joe Shmoe’s department as Sales but that didn’t list Joe Shmoe among the employees in the Sales department would be inconsistent.
It’s a logical property of the DB, independent of the actual data.
Integrity:
A DB that reported jOe SaleS to be a member of the Shmoe department would lack integrity.
jOe SaleS isn’t a valid employee name and Shmoe isn’t a valid department.
That’s not logically invalid, but it is invalid relative to the rules that govern data content.