I haven't been able to fully grasp the differences. Can you describe both concepts and use real world examples?
An identifying relationship is when the existence of a row in a child table depends on a row in a parent table. This may be confusing because it's common practice these days to create a pseudokey for a child table, but not make the foreign key to the parent part of the child's primary key. Formally, the "right" way to do this is to make the foreign key part of the child's primary key. But the logical relationship is that the child cannot exist without the parent.
Example: A Person has one or more phone numbers. If they had just one phone number, we could simply store it in a column of Person. Since we want to support multiple phone numbers, we make a second table PhoneNumbers, whose primary key includes the person_id referencing the Person table.
We may think of the phone number(s) as belonging to a person, even though they are modeled as attributes of a separate table. This is a strong clue that this is an identifying relationship (even if we don't literally include person_id in the primary key of PhoneNumbers).
A non-identifying relationship is when the primary key attributes of the parent must not become primary key attributes of the child. A good example of this is a lookup table, such as a foreign key on Person.state referencing the primary key of States.state. Person is a child table with respect to States. But a row in Person is not identified by its state attribute. I.e. state is not part of the primary key of Person.
A non-identifying relationship can be optional or mandatory, which means the foreign key column allows NULL or disallows NULL, respectively.
See also my answer to Still Confused About Identifying vs. Non-Identifying Relationships