What is the difference between the Eastern and Western views of the one essence/substance of the Godhead? The philosophical metaphysics used by the Eastern and Western church's seem to be heavily influenced by Plato and Aristotle, respectively. How does this affect each's theology of the substance/essence in the Godhead?
The divine energy of Eastern Orthodox theology defines the divine 'essence' as an abstract definition of what 'God is', while the 'energy' is the ‘working-energy’ of what God does.  The western theological meaning of ‘essence’ includes ‘working energy’ in the definition already. For anybody familiar with programming terms, Eastern Orthodox sort of splits God into a class (essence) and eternally substantiated object of that class (energy) both class and object are eternal and uncreated. 

For some reason the eastern Monk Gregory Palamas (1296–1359) came up with a distinction between God’s Essence and God’s energy.  The Essence-Energy distinction was a theological framework used by Palamas to explain how creation can participate in the experience of God, without becoming God. The idea in simple form is that ‘the manifestation of God’ or ‘the experience of God’ must be separated out from the non-experience-able essence of God; otherwise we are left with a Pantheistic view. This Pantheistic view would be seen as Neo-Platonic--Aristotelian. Something the western view is accused of being.

From the Western standpoint, this distinction was never needed as Western theologians simply used current handy Greek terms to explain the problem away in accordance with Christian concepts of God. The use of these terms does not automatically make the western view of God to be like Plato’s or Aristotle’s just because words have been borrowed. More specifically Thomas Aquinas used concepts such as Aristotle’s distinction between ‘Potentiality versus Actuality’ as well as ‘Actus Purus’, to show that to the Christian view of God automatically avoided Pantheism.

The idiots guide (i.e. my own) to Aristotle’s view of what ‘essence’ is and how it was borrowed and modified under a Christian lens is pretty simple. There is a kind of ‘essence’ beyond the particular embodiment, for example, if there is something called human, then Frank is only an example of, human essence, In addition if Frank is to really be a real human (his potential) he needs to become one actually through an chain of events (working-energy). Applying these terms to God necessarily means that God can’t have any such division of potential versus actualization of potential.  God is perfect he can’t become actualized and so his essence and working-energy is within the same incomprehensible perfection of his singular being.

To apply this concept to the eastern proposal, a division of essence and energy is rejected. Since God can’t have potential that is not actualized, and actualization implies working-energy, the idea of a separating energy from essence is incongruous to God’s perfection. Manifestations of his presence in time and space are not based on any separation within his own perfections and this does not allow for any kind of Pantheistic view by default. 

I have oversimplified all the lengthy philosophical terms a bit, but I think you get the western idea from this. Hopefully you also sense the somewhat nauseous headache one obtains splicing these terms and how needless they are when approaching the more simple faith of those in the New Testament.

So what am I saying, well both camps seem to just use different terms to try and say the same thing. They both resorted to the overly intellectual use of philosophical distinctions, which was probably partly required as a reaction to on the multitude of heresies over the Trinity, etc., which the church suffered at that time faced. However if Palamas meant that the uncreated energies were truly ontologically distinct from God;s essence ,as are the three persons of the Trinity, then Palamas is less innocent that it would appear under my western view.  Some scholars say he did make this true distinction and other say he did not.

For a more complicated review these two articles basically throw in all the controverted points:

http://en.wikipedia.org/wiki/Potentiality_and_actuality#Potentiality

http://en.wikipedia.org/wiki/Essence-Energies_distinction