Thank you for your question regarding earthing systems and lightning protection systems, it is our pleasure to help.
Lighting protection systems are an above-grade conductor network designed along the lines of a Faraday Cage, that is used to protect a structure from the effects of lightning discharges (direct or indirect).
An earthing system is a below-grade conductor network designed to allow electrical energy (from an above-grade network) to transition from the conductor network into the surrounding native soil (earth). Many different above-grade conductor networks rely on the earthing system for the discharge of unwanted electrical energies. This includes the lightning protection system, but also includes the main electrical ground network, the building structural steel, telephone and internet grounding systems, computer server room grounding systems, static grounding systems, gas pipe line ground systems, water pipe grounding, and many more.
So the simplified relationship between the lightning protection system (LPS) and the earthing system is that the LPS catches the electrical energy and routes it away from the structure and down to the earthing system which then dissipates the electrical energy into the earth.
We hope this answers your question. Please feel free to contact us again should you have any further questions in the future.
The Engineering Team at E&S Grounding Solutions
Photo credit: http://www.flickr.com/photos/colinedwards/291628090/sizes/z/in/photostream/