Significant can mean special and noteworthy. It is also used in mathematics to denote the number of digits after the decimal point-
Important can mean high-ranking.
Sally is my significant other.
- While Sally is important she would go beyond that. She would be special.
Find the value of pi to four significant digits. Answer is 3.1415
- The value of pi goes on forever, so we must choose to stop at a certain number of digits after the decimal point.
The president of the club is my friend.
- The president is a high-ranking position within an organization in addition to other leaders. Normal members of the organization would not have the high-ranking positions, so they would not hold an important position.