Time zones don't normally affect a person's every day life, as long as one remains within the same time zone. However, there are some instances where time zones can have a cultural, social, or in some cases even a political impact.
Let us first of all look at the cultural impact. You could use New Year's Eve as an example, which is a cultural tradition celebrated all across the world, in all different time zones. Here the cultural effect is clear: whilst most people across the world celebrate New Year's Eve, often with firework displays and other festivities, it is impossible for all of these people to celebrate the event at the same time. For example, people on the Pacific island of Tonga are the first ones to celebrate the New Year when their clocks strike midnight. However, (ignoring the uninhabited islands Baker Island and Howland Island), American Samoa is the last inhabited country to celebrate the New Year over a day later.
On a social level, you could argue that the time zones affect people who wish to communicate with people in a different time zone. For example, New Zealand is 13 hours ahead of the UK. This means that if a person were to pick up the phone at 11am in the UK to give their friend in New Zealand a spontaneous call, they would very like wake their friends up, given that it would be midnight in New Zealand. This makes social interaction a little bit more complicated, as it requires careful forward planning.
Politically, different time zones can make it difficult for important political news to reach all people across the world at the same time. For example, when the first airplane crashed into the North Tower of the World Trade Center at 8:46am local time, it would have been 1:46am the next day in Wellington, New Zealand, meaning that most people wouldn't have heard these terrible news until about at least 5 hours later. Therefore, the release of political news that impacts people across several time zones has got to be planned with this possible delay in mind.