There is considerable overlap and some specific doctors have even more overlap than others. In broad generalities, OMFS (who are fully dental trained) deal more with the teeth and maxilofacial skeletal system. ENT typically handle nasal and sinus issues; laryngeal and voice issues; tonsil problems; neck soft tissue problems such as head and neck cancers, enlarged lymph nodes, etc.
If a patient has tonsillitis, no swelling, but coughing up tonsil stones which are causing halitosis that can be smelled from feet away, would tonsillectomy solve this problem?