What defines clinical versus diagnostic audiometer?

You might consider expanding this question to include screening audiometer, alongside diagnostic and clinical audiometers. The functionality and sophistication increases along a spectrum from screening to clinical audiometers. To some extent the distinction between different types of audiometers in the modern range of audiometers is blurred as there is a great deal of overlap, but in addition to features, certain other factors influence the classification such as the frequency accuracy required and the range of hearing levels that can be tested. 

Committees such as ANSI and IEC have published such as definitions for the minimum number of features required to meet a classification. The standards are in close agreement.

For example IEC 60645-1 (2017) outlines four types of audiometer (Type 1 Advanced clinical/research; Type 2 Clinical; Type 3 Basic Diagnostic; Type 4 Screening)  and provides a breakdown of the various features that apply for each type.

References and caveats
IEC 60645-1 (2017) Electroacoustics - Audiometric equipment - Part 1: Equipment for pure-tone and speech audiometry