There's a base assumption at play here that makes the addressing the issue at hand much more murkier than it should be.
Psychiatry sees all mental health problems as, in root, organic in nature. In other words, there is a chemical imbalance, a brain trauma, or a genetic component that creates the symptoms. These mental health issues can be seen as "cured" through medical regimes, but, many other illnesses, considered under remission, since a chemical imbalance caused by a genetic component cannot be "cured". The DSM (V is the latest incarnation) uses symptomatic observations as base criteria - not necessarily biological markers, but medical therapy is based on biomarkers (for example, a regulation in serotonin uptake). While this is a gross oversimplification of the matter, it paints a general picture of what happens with the organic position of mental illnesses. In a very simple word, psychiatry views mental illnesses as a nature problem.
Psychology, on the other hand, does not see all "mental illnesses" (as defined in the DSM) as organic in nature. As swillden mentioned, psychotherapy (of which there are many intervention methods) assist in managing the situation. Much of it takes root in mindfulness - not only in a social perspective, but also a reframing and re-internalization of current and past events. Others might take a family based approach, not only in dealing with the specific issues the primary client is dealing with, but also how their immediate social structure responds to their condition. CBT (Cognitive Behavioral Therapy) *might* go a step further and integrate neuroscience to determine if there is a biological component to their client's condition (but this tendency is still far and few between, considering the cost, the protocols required, and the length of time). Depending of the epistemological perspective of psychology, psychologists don't "cure", because the client isn't sick, they are maladjusted (through previous rationalizations or emotional internalizations of events and situations). Using a similar oversimplification, psychology sees many mental illnesses (note: not all) as a nurture problem (a learning/behavioral problem that has both an individual and social/cultural component).
There is a grey point in between these two, apparently competing points of view, which come up often in these discussions. What happens with schizophrenia? Or with a catatonic patient? A medical regime may assist in managing the symptoms, but without some measure of psychotherapy, the person will have a much harder time dealing with their inner situation. I doubt that something like psychoanalysis (or tools from psychodynamics) will work well, but perhaps a cognitive behavioral intervention might have a better success rate. Or even some of the tools from the Humanistic school of thought can help.