Why is there a structural deficit/surplus?
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.