The elevator trim mainly acts like a cruise control. Once the aircraft is trimmed for a certain airspeed, it will tend to maintain that airspeed regardless of the power setting.
One common misconception among pilots is that the trim keeps the current altitude or flight path. In reality, it is the trimmed airspeed the aircraft tends to maintain.
When you change power without changing the elevator trim, the aircraft will adjust its vertical speed in order to stay at approximately the same airspeed.
Increase power → the aircraft tends to climb.
Reduce power → the aircraft tends to descend.
When you want to change the airspeed, you need to change the trim as well.
Remember: trim for airspeed!