Why is there a structural deficit/surplus? I'm going through Fiscal Policy and I'm reading about Structural Budget and Cyclical Budget. To my understanding, Cyclical Budget is what occurs when automatic fiscal policy is in use. So G and T are equal at full employment, if the economy falls into a recession, the automatic fiscal policy will trigger a cyclical deficit. 

What I don't understand is the point of Structural Budget. Why is it run if the economy will return to full employment in the long run automatically? I understand that it gives an extra push but why don't we just let it handle it automatically?
In terms of rule's logic, they are the same. The logic of structural budget is to provide stability and long-term planning to government spending. This is implemented by first finding the "long-run" revenue levels. Long run here means the economy in its potential. Additionally, for countries which rely a lot on some particular industries like natural resources (e.g. Copper in Chile), an estimation of the "long-run" price of such resources is required. These estimates are usually conducted by an independent, non-partisan, expert-based body, not affiliated with the government.

Once these revenues are defined, then a rule is chosen. It could be a balanced budget (all long-run revenues are spent), which is what you are calling Cyclical Budget, or a deficit/surplus, defined as % of total revenues, which is what you seem to refer to as Structural budget deficit/surplus. Either of these methods provide stability to spending, as the committee evaluating the long-term revenues of the government does this for several years in advance. 

Why to chose a structural deficit/surplus? A structural surplus can be chosen to accumulate savings over time, and have a cushion in time of a crisis. In the case of Chile, the aim was to build a cushion in case the long-run price of Copper fell sharply, and long-term government revenues shrank considerably. Others might chose a deficit in order to provide a constant stimulus to the economy. Naturally, the size of the public debt is something to have in consideration regarding which rule to choose.

You can read a bit more about this rule, for the case of Chile, here or here.