Amazingly, they lived in the same era, and are believed to have met each other. By all accounts, they did not get on all that well.
Confucius

He worked as a teacher, and he would teach any committed student, regardless of their social background or status. He taught from the classic literature of the past, often singing verses from ancient texts, accompanying himself on the zither (an early type of guitar.)
He argued, particularly in his famous book The Analects, that rulers should be selected on merit, rather than inheritance, and that they should have devotion for their subjects, and that they should be of virtuous character to earn the respect of the people. The subject has a duty to obey the ruler, but the ruler also has a duty to listen to criticism.
He became active in politics and he was promoted to be the Minister of Justice. Under the reforms he introduced, it is said that crime was virtually eliminated.
Lao Tzu

Lao Tzu maintained that thinking is the cause of all our problems. He advocated that we should avoid the pursuit of goals and just let the eternal Tao flow through us. This he believed to be the skilful approach to life, known as wu-wei, meaning non-effort, or non-striving. This would put us all into harmony with the universe.
He is reputed to have said, "He who speaks does not know, he who knows does not speak," and "The Tao that can be talked about is not the eternal Tao."
When he was aged 80, he became exasperated with the ways of men, and disappeared into the western desert, never to return.
So here we have two different suggestions to solve the problems of society.
Confucius: Everyone should concentrate on preserving the rituals and customs of the past, with good manners and proper etiquette.
Lao Tzu: Everyone should do whatever they want, not trying too hard, but just going along with the flow.
If you were a Chinese philosopher, would you be Confucius, or would you be Lao Tzu?