My psychiatrist and my therapist have given me different diagnoses but I'm not terribly concerned about that. None of my diagnoses are "significant" (just standard mood and anxiety disorders) and I feel adequately treated by both providers.

Now, if either had given me a diagnosis with significant implications (e.g. a personality disorder or schizophrenia), I would follow up on that. I would also follow up if the two diagnoses were at odds with each other. For example, a friend of mine was diagnosed with generalized anxiety disorder by one provider, and ADHD masking as anxiety by another provider. Since those diagnoses have "opposing" treatment strategies (e.g. stimulants for ADHD, benzos for anxiety from a medication perspective), he needed to get that resolved before comitting to a course of treatment.