The original answer, whatever it was, has been removed. I'm not sure what the content was, but here's a new answer.

Sexagesimal number systems--counting systems based on 60--are very useful in societies that rely on mental counting because the number 60, like the number 12, is a highly divisible number. It can be divided by 1, 2, 3, 4, 5, 6, 10, 12, 15 and 30. This means it was popular among earlier societies, such as the Sumerians and then Babylonians, whose counting system was based in 60s. (The Chinese had a a decimal system).

This system was adopted for measurement of the earth by various Greeks and was codified by Ptolemy in the Almagest into dividing the Earth into 360 degrees around, each degree divided into minutes and 'second' minutes, a system we still use when we write degrees of latitude. It's just very easy to work with.

As for minutes of time: hours are divided into sixty minutes and minutes into sixty seconds for the reasons described above: decimals systems were not as prevalent and the possibilities of 60 and 360 were widely known and accepted. As clocks became more developed and precise and more finely measured time spread into the general populations every day experiences, minutes and seconds appeared on the clock. The original, easily divisible numbers prevailed because they are easily divisible so many useful ways.

Which is not to say that an attempt hasn't been made to decimalize time in recent history. Following the French Revolution, France decimalized their measurements, including linear measurement (e.g. the metre) and time (weeks of ten days, days of ten hours etc.). The complete decimalization was introduced but only endured around a decade. However, the decimalized linear, weight and volume measurement (e.g. the introduction of the metre, divided into tens and hundreds and thousands of parts), persisted and was later adopted as an international technical standard. Similarly, as international time was being established towards the end of the 19th century (travel and telegraphs making this necessary and possible), there was a push by the French to re-introduce a decimal hour and minute system (although not a decimal day), but it didn't prevail as a standard.

For the discussion of decimal time at the end of the 19th century, see Einstein's Clocks, Poincare's Maps - Peter Galison