A Chinese teacher may be one of the 2 categories:
Teacher of Chinese
The technical differences are extremely small and yet valuable when making your decision. Chinese teacher teaches Chinese to Chinese speakers, when a teacher of Chinese teaches Chinese as a second or a foreign language.
Unlike a Chinese teacher (or a teacher of Chinese) a tutor usually doesn't have a college or university degree in teaching, education or linguistics.
Tutors often use unconventional and more cultural methods of teaching. tutors tend to focus on student's experience and language natural development rather than typical books.
A teacher can teach you grammar, structures, time tenses and a tutor can help you expand your vocabulary and dip your toes into culture, however...
Practicing with Chinese Native Speakers helps you improve on pronunciation and understanding of a fast, real, dialectic speech, which is so different in different parts of China.