Difference between greatest and highest

Hi Tharmacy,
In a very simplified way, when you speak of greatest think of it as big in a horizontal way and highest as big in a vertical way, i.e. greatest includes more aspects of the object, while highest aspires to be more pure in one aspect.

Think of greatest as that which is related to size or magnitude.
Think of highest in terms of quantifiable numbers...altitude or most exalted.

Example:
I get the greatest pain in the arse from my annoying children's bad behavior.

I have the highest respect and admiration for well-behaved children.