Adapted from an earlier answer:

Hipparchus of Nicaea (c. ~190-120 BCE) seems to have been the first person to propose using a grid system to find the position of cities (and other places) on a globe, which implies an understanding of longitude. He built on earlier work by Eratosthenes of Cyrene (c. 276-194 BCE), who had mapped the known Earth, including finding its circumference. Hipparchus' method of finding the longitude of places was to use the differences in timing of lunar eclipses at different points on the globe to calculate the difference between local time of those points; the drawback is that there was no accurate-enough method of timekeeping to lead to useful calculations. The difference in local time between observed beginning and end of the eclipses would serve, essentially, as a way to understand the longitude between places.

The knowledge that local time would be different at different points on the globe is what led eventually to what's called "the discovery of the longitude" in around 1760 or so, when two methods of reliably finding longitude using time were discovered and implemented.

The most straightforward way of determining longitude is comparing local time to time elsewhere on the globe (usually, Greenwich Mean Time) and figuring out the time difference; if it's 1 p.m. at Greenwich at your local noon, you're 15 degrees west of Greenwich. Once marine chronometers became widespread, the longitude problem was easier to solve; but, chronometers were only provided to British naval ships traveling in far distant waters starting in the 1790s, and did not become standard issue until the 1840s. (Captains or masters could buy chronometers, although they were horribly expensive -- 60 to 100 guineas new, plus 5 or 10 per year for cleaning/resetting, and ships needed three to correct for errors.)

You could also calculate your time, and thus position, based on "lunar distances" -- either finding the degrees between the moon and another celestial body, or by measuring the positions of the moons of Jupiter, to compare it to tables and find Greenwich mean time. I am way in over my head on the math on these, but [Wiki has](https://en.wikipedia.org/wiki/Lunar_distance_(navigation) what I am told is a perfectly cromulent summary.

Regarding "the discovery of the clock," I'm not entirely sure what you're talking about. People have used various implements to measure time since antiquity -- sundials and water clocks come to mind, as do hourglasses and their variants, but it's also quite possible to estimate time by looking at the position of the sun in the sky, and starting and ending a "day" from local noon. In Europe, religious orders divided the day into hours based on times of prayer, well before mechanical clocks were developed in the 1300s or so. But to answer your basic question, yes, people knew in antiquity that time was different in different parts of the globe.