Have you ever said "let's hang out Tuesday afternoon"? Abolishing time zones makes that a useless phrase for a large slice of the world - consider the area where the date changes in the middle of the afternoon. And which days are the weekdays? The ones where it's Monday-Friday in the morning (with Tuesday-Saturday afternoons) or where it's Monday-Friday in the afternoon (with Sunday-Thursday mornings)?

This doesn't really solve the problems with times, it just moves them from times to dates.