It's not really about the speed of the clock, but the gravitational pull, or in fact its position on the gravitational well compared to the other clock that causes the difference, at least how I understand it anyway.

I hope someone can correct me if I'm wrong in saying this, but I think that if the two clocks started on top of a mountain, one was taken and kept at the same distance from the centre of the earth as it circumvented the globe, assuming no other gravitational pulls ( the sun, the moon etc.) then when the travelling clock has returned, they should still be at the same time irrespective of the speed it has travelled...

Erm, I'd like to help but I'm not quite sure what you mean about "the relative nature of the speeds".

Special relativity states that, amongst other things, when objects travel at a speed, time slows down for that object, i.e. the clock ticks at a lesser rate than one that is stationary. Although this change is only really observed at speeds near the speed of light, it can just be noticed by flying an aeroplane around the earth with an atomic clock in it. This is the phenomenon you've described in the title.

What kekatev is describing is an effect predicted by general relativity, which states that time slows down the lower you are in a gravitational field. If a clock is held at the same height above the earth and moved along (as stated in your situation) general relativity does not predict a change in the time. However, special relativity does.

I hope this helps, and if you could rephrase what you wrote, i'd be happy to help,