One minor motivation to develop really precise clocks, accurate to 1 part in 10**18 (Ytterbium lattice clocks, for one example https://en.wikipedia.org/wiki/Atomic_clock ) is that it is possible that two different such clocks (each using a different type of atom), a few meters apart on a desktop, could show a variation in rate over a reasonable period of time, say a month or a year.
Put more simply, suppose you could compare the frequency of two such clocks, with different types of atoms, and express that result to a precision of one part in 10**18. A month or a year later, you do the same comparison, you might discover that the ratio of frequencies are different by a few parts in 10**18, or perhaps much larger.
This would be a laboratory verification of a recent astronomical observation, a one-part-in-100,000 change over a period of 6 billion years: http://www.physicscentral.com/explore/action/constant.cfm If that rate of change were constant, it would be one part in 600 trillion per year. If an equivalent laboratory variation rate could be compared to a precision of 10**18, that would be 1600 units per year, or 133 per month, or perhaps 4 per day. ×