You will find happiness in satisfaction while you may not find satisfaction in happiness. However, happiness can be defined as a state of the emotion and mind being elated at an occurring event. On the other hand, satisfaction can be defined as a state of one's desire is met. Happiness can be by choice, that is, you can choose to be happy. Even the poor who seem not to be satisfied is happy sometimes.
Nonetheless, you have to strive to find satisfaction. Happiness is a state of emotional reaction to physical occurrences, whereby satisfaction is purely a state of physical occurrences. That is to say, for happiness, your mind can control its reaction to a physical occurrence, but for satisfaction, a particular physical occurrence will control the mind. There is contentment in happiness, but there might not be contentment in satisfaction. So, one's personal belief has a long role to play in determining your happiness and satisfaction.