“Therapist” is a vague term. Anyone doing therapy with you is your therapist. (More specifically, anyone doing psychotherapy is your psychotherapist. There are also physical therapists, of course, but I assume you aren’t asking about those.) “Counselor” is even more general. Anyone helping you with just about anything can legitimately be called your counselor. States will have specific laws about who can hang out a shingle calling themselves a “counselor” or “therapist” but in many states it’s absolutely anybody. That is certainly the case in my state of Colorado, although you have to register as an “unlicensed psychotherapist” so the state can record complaints against you. (That part is unique to Colorado, I think.)
Most (all?) states also have one or more specific classifications for mid-level practitioners who have been licensed by the state. Colorado has “Licensed Practical Counselors” (LPCs). I believe the equivalent in most states is “Licensed Family and Marriage Counselor” or “Marriage, Family and Child Counselor.” These require a Master’s Degree, usually in Counseling, in addition to the state’s licensing requirements. I believe all states also recognize counseling and psychotherapy to fall within the scope of practice of a Licensed Clinical Social Worker (LCSW). A Master of Social Work is a prerequisite for becoming an LCSW. I believe LCSWs are more commonly accepted by insurance companies and other payers and are generally regarded as having a slightly higher qualification than LPC/FMCs.
To further complicate matters, there are also Certified Addictions Counselors, who are licensed to do drug and alcohol counseling. In Colorado, the lowest level of certification, CAC-I, doesn’t even require a bachelor’s degree. (But many have Master’s degrees.)
Most or all states require someone licensed or working as a psychologist to have a PhD in Psychology or a PsyD (Doctor of Psychology). Psychologists are very highly trained (but of course may still be crap at doing psychotherapy.) Psychologists are often trained primarily as scientists, and may only do research and little or no counseling of patients. Others work primarily as psychotherapists and have done little or no research, even in school. Most fall somewhere in between. Besides psychotherapy, many clinical psychologists perform psychological evaluations and other testing, which can only be performed by a psychologist, so an LCSW, LFMC, or MFCC may refer you to a psychologist if they want some specific testing done.
A psychiatrist is a medical doctor who, following completion of their MD, performed a residency in psychiatry. Most psychiatrists specialize in prescribing psychiatric medications, and if you see a psychiatrist it will almost always be because a family doctor or a therapist (of whatever qualification) referred you. Some psychiatrists also do psychotherapy, however.
There isn’t much difference in the psychotherapy done by each type of practitioner, although each practitioner will have their own style and may specialize in certain types of afflictions, and their educational background may influence how they see various mental health issues. Also, each one should have additional knowledge and skills beyond psychotherapy that may be useful. An LCSW, for example, should have at least some knowledge of how to work with schools to deal with emotional or psychological problems, while a psychiatrist should be skilled at recognizing other medical conditions such as autism or thyroid problems, which may affect mood and behavior.