Generic questionnaires assess concepts that are relevant to everyone. The results enable comparison of outcomes between different samples (e.g. comparison with the general population or with people living with another chronic condition).
However some items may be irrelevant for people with diabetes, or may bias the results. For example many depression measures include items about energy levels, which for people with diabetes may be influenced by high blood glucose levels and not necessarily indicate a symptom of depression. On the other hand, relevant domains are not covered by these items, for example, the impact of diabetes on dietary freedom.
Overall, a diabetes-specific instrument is preferred. The items are more relevant for people with diabetes, which will make the measure more meaningful, acceptable and less burdensome to complete. These measures are more likely to be sensitive to change, which is important when evaluating the effect of an intervention. Therefore, the results of these diabetes-specific measures enhance the understanding of the impact of diabetes. That being said, the validity of the results will depend on the quality of the measure and its overall fit with the aims of your project.