An urgent task needs to be done soon. An important task needs to be done by you.

1) If it's a work task and you're at the bottom of a hierarchy, everything given to you by the boss is 'important' by this definition.

2) It may be important to them - the trick is finding the right person.

Taking your kid to a scary dentist appointment is urgent and important; it can't be delayed and it can't be anyone else.