What is the difference between
"clinical" and "chronic" depression?

The term "clinical" as an adjective or qualifier of mental and emotional disorders is not currently used in the Diagnostic and Statistical Manual (DSM) as a valid descriptor.  It probably came into use to distinguish disorders that met the profession's criteria for symptoms warranting intervention from those of a less serious nature.  It probably was convenient to use when dealing with the public and press rather than give a more accurate and detailed description that would only mean something to those in the mental health field.  Various types of depression are diagnosed with several descriptors some of which consist of a series of detailed sub-descriptors strung together.  An example of this is "Severe Major Depressive Disorder With Psychotic Features In Partial Remission With Postpartum Onset."  As you can see, this can prove to be a bit unwieldy for most folks (including diagnosticians) so the use of a single term such as "clinical" to denote any of the diagnosable types of depression would naturally catch on very quickly.

When diagnosing disorders, one question to be answered is how long have the symptoms been evident?  That answer tells us whether it is "chronic" or "acute" in nature.  The category of "chronic" might be used to describe a specific disorder with symptoms that have been evident in the patient's life for at least six months, while "acute" might be used when symptoms are less than six months old.  For another disorder, the times might be two years for chronic, and for another disorder 30 days might qualify for acute.  There is no hard and fast rule that I know of for determining when to use chronic or acute as a descriptor except that chronic refers to a greater duration of time than does acute for any given situation.  Proper use is determined by consulting an industry standard such as the DSM for each specific disorder in question.