Postpone: do something at a different time
Extend: keep doing something for a longer time
Examples:
Can we postpone the meeting to tomorrow?
We extended the deadline because this project is taking more time than we thought.
The race was postponed to next week due to the weather.
The store extended their hours to accommodate people who get out of work late.
The following are how the words are generally used.
To postpone is to stop until a later time. If you plan to go on a trip this weekend but something unexpected comes up, you postpone the trip to the next weekend.
To extend is to take an event already in progress and continue it longer than expected. Again with the trip, if you plan to stay for 2 days but an opportunity comes up so you can stay for 3 more days and you do that is extending the trip.
Postpone is arbitrary, an action initiated by someone who has the authority to lag an existing plan.
Extend means to push a stated deadline further into the future.