Background: The purpose of this study was to assess and compare the prevalence of self-reported mental disorders, psychological distress, and psychotropic drug consumption among people with diabetes mellitus (DM) and matched non-DM controls. Methods: A case–controlled study using data from the Spanish National Health Interview Survey was conducted in 2017. We identified 2116 DM adults (aged ≥35 years). Non-DM controls were matched 1:1 by age, sex, and province of residence. Results: Prevalence of mental disorders (25.0% vs. 19.2%; p < 0.001), psychological distress (29% vs. 19.5%; p < 0.001), and consumption of psychiatric medications (29.7% vs. 23.5%; p < 0.001) among DM cases were higher than those among matched non-DM controls. The DM patient variables associated with experiencing a mental disorder, psychological distress, and consumption of psychiatric medications were: being a woman, worse self-rated health, and a visit to a psychologist within the last year. Older age (≥80 years) was associated with a lower probability of reporting mental disorders and psychological distress among DM cases. Not practicing physical exercise was significantly associated with experiencing psychological distress. Conclusions: Adults with DM included in our investigation have a significantly higher prevalence of mental disorders, psychological distress, and consumption of psychiatric medications than non-DM controls. It is necessary to implement screening strategies and psychological interventions to improve the mental health of DM patients in Spain, focusing especially on women and those aged 35 to 59 years.