What is relationship between mental health and mental hygiene?

Same: Mental hygiene is an older term for mental health, but the terms can be used interchangeably. The mental hygiene movement took off in the early 1900's, and was basically concerned with the improvement of the care of people with mental disorders.

Mental Health: Refers basically to our state of emotional wellness where as mental hygiene is about how we take care of ourself emotionally.

MH/MHygiene Related: Mental Health is often used to refer to a state of physical, mental, social and spiritual wellbeing of the individual. MH is one of the components of the broad concept of health concerned with an optimum level of emotional and behavioural adjustment of the individual. Mental hygiene is a Science which deals with the process of attaining mental health and preserving mental health in the society. The term mental health is closely related with the term mental hygiene as the main objective of mental hygiene is to attain mental health. In other words, mental hygiene is a means of mental health. That is why we can say that mental hygiene is the means and mental health is the end.