Exactly what do you mean by "assessments?" I know LMHCs who do psychosocial assessments or assessments to determine whether or not someone needs to be hospitalized (emergency services). They do not do psychological testing, which is a bit different.
LMHCs provide counseling/therapy, including psychodynamic and cognitive-behavioral therapy. Counseling psychologists have doctoral degrees and can do clinical work but also can conduct research and teach at the college level, whereas LMHCs are master's level clinicians who can practice therapy independently but don't run their own research lab or teach at the college level (except for some adjunct/clinical professors -- not tenure-tracked).
Other posters: Please correct me if I'm wrong in anything I've written. I work with LMHCs now and I'm posting based on what I know about their work.
OP: I hope I've answered your question.