There is at least one way to use "тут" when it is not interchangeable with здесь. It can be used to refer to the moment of time immediately following the action described before. In this sense "тут" can be translated as "suddenly" or "immediately".

Normally здесь is used to describe the position in space only, not in time.

Quote Originally Posted by Zaya
Ну вот сам дальше и объясняй.