Psychiatrists diagnose your disorder by symptoms and functionality, that is how debilitating your anxiety/panic disorder is. Their method of treatment is meds and suggest you seek therapy in addition to their treatment for maximum results.
Many psychologists (depending on country/state) do not prescribe medications. They deal with your anxiety/panic with exposure therapies via communication. They tend to listen to your 'story' more than a psychiatrist because that is their mode of healing you.
The biggest difference IMO is that many that have anxiety/panic disorder have chemical imbalances in the brain from past traumas and those imbalances must be balanced before you can begin the mental trauma healing. And that must be done with meds.
Physiological problems can cause anxiety/panic (thyroid disorder, adrenal disorder, etc.) and that type must be dealt with by a specialist before psychological healing can take place.
Don't get me wrong, psychologists have a place. Their therapy is proven effective (and you do not have to go to a psychologist for therapy; there are cheaper licensed therapists available that do it all; CBT, hypnotherapy, EMDR, etc.). But, once again IMO, they are helpful for minor to moderate anxieties or in addition to a psychiatrist.
So, minor to moderate anxiety/panic, psychologist. Debilitating anxiety/panic, psychiatrist, therapist, exercise, healthy diet, lifestyle change.
Just my 2 cents.