A favorable moment or turning point that can be seized to start or change something. Often used with ～を契機に or ～を契機として.
Studying abroad became an opportunity that changed his outlook on life.
I want to use this success as a springboard to expand the business further.
The direct cause or catalyst that sets an event in motion. Often used with ～が契機で or ～が契機となって.
That incident was the trigger for the law to be revised.
A small mistake became the cause that escalated into a major problem.
I used the move as an opportunity to reconsider my daily habits.
His illness was the trigger for him to quit smoking.
The accident became the trigger for strengthened safety measures.
Taking this failure as a turning point, he started studying seriously.
転機 refers to a turning point in one's life or a situation, whereas 契機 emphasizes the specific event that serves as the catalyst for that turn.