Mental health professionals aren't as different as specialist doctors, but they each have specific roles that don't always overlap. I'm going to give you a primer on the most common kinds so you know who does what when you go in for therapy (which I highly recommend you do from time to time even if you don't have a mental illness.)
Psychiatrists have gone to medical school, so they have a medical degree and can prescribe medication. Generally, an appointment with a psychiatrist is a 15-minute med check centered around any symptoms, side effects, and prescriptions. Psychiatrists usually don't conduct psychotherapy because the schooling required to become a medical doctor and psychotherapist isn't feasible for a vast majority of people.
2. Psychiatric/mental health nurse practitioners
Psychiatric NPs are the only other mental health clinicians qualified to prescribe medication. Like other nurse practitioners, these clinicians have at least a master's degree, if not a PhD, with a specialization in psychiatry. They can conduct assessments and diagnoses for mental illness/substance abuse disorders and provide some therapy depending on their level of training and certification. Nurses may also works as case managers or patient advocates.
Psychologists are the most versed in psychotherapy because they must have competed at least a PhD or PsyD in psychology, a 2-3 year internship to further train in treatment methods, therapy, etc., and then they can apply for licensure. (I have a long road ahead of me, I know.) They can conduct assessments and diagnoses for mental illnesses/substance abuse disorders, but they can NOT prescribe medication.
Psychologists come in all kinds of flavors too. You have health psychologists, clinical psychologists, counseling psychologists, forensic psychologists, and many more. Some of these roles are more research-oriented while others are more practice-oriented, and some are somewhere in between. The ones you're most likely to encounter would be clinical and counseling psychologists.
Clinical psychologists focus on treating mental, emotional, and behavioral disorders. They will usually specialize in certain mental illnesses like anxiety or depression or chronic illnesses like diabetes or obesity. Some will work only with children so they would be a child psychologist, but those who work with adults are just called clinical psychologists.
Counseling psychologists also deal with disorders, but they take less severe cases than clinical psychologists. They can also help people cope with struggles in life such as the death of a loved one, career issues, personal growth problems, and so much more.
I want to underpin this and say that clinical psychologists can also help with personal growth problems too. I happen to see one at my university's counseling center for that exact reason, so you can still see a clinical psychologist even if you're not mentally ill.
4. Counselors, clinicians, therapists
These folks carry Master's degree in a mental health-related field and are trained to evaluate a person's mental health and provide certain therapeutic techniques. They can NOT conduct assessments like psychologists and psychiatric nurse practitioners. These folks come in all flavors too, but the following three are pretty common:
Licensed Professional Counselor (LPC)
Marriage and Family Therapist (MFT)
Licensed Clinical Alcohol & Drug Abuse Counselor (LCADAC)
5. Licensed clinical social workers (LCSW)
Social workers have a Master's degree in social work and can perform as many functions as there are colors. Except for the ones they're not qualified for. Like counselors and therapists, social workers can't conduct assessments. They can provide any therapeutic techniques they're trained in, work as case managers or advocates, and so on.
6. Primary care physician (PCP) or Family nurse practitioner (FNP)
This one is the least common scenario, but occasionally people will choose to receive psychoactive medication from their PCP or FNP without pursuing therapy. Because doctors have a medical degree, they're technically qualified to do this, but are understandably less inclined to do so than psychiatrists. Also, mental illnesses are usually best managed with a combination of psychotherapy and medication.
So now you hopefully understand why you need to know who to ask for next time you step in your university's counseling center, your local hospital's psychiatry department, a local private practice, or whatnot. Most mental health clinicians can't prescribe medication, but some can, and not all psychologists are focused entirely on treating mental illness.