Practical Example:
I want to talk with a company in Florida and need to schedule a meeting. They tell me to call at 17:00. Now I need to translate that into my local time. So I look up the time zones. Florida is UTC -5, while Germany is UTC +1, meaning I'm 6 hours ahead and I would have to call them at 23:00 German time. This is obviously bad for me, so I try to reschedule. They tell me their office times are from 7:00 to 18:00. So now, after some pointless arithmetic, we settle on a compromise at 10:00 Florida time, which is 16:00 German time.
So I call them at 16:00 and reach no one in the conference call and are annoyed that they left me hanging. But slightly over an hour alter, I get email why I didn't show up for the meeting.
Fun Fact: The westernmost part of Florida is actually at UTC -6, so both of us showed up on the "right" time.
No-Timezone Example:
"Let's do the call at 17:00."
"That's in the middle of the night for me. What's you office hours?"
"They are 7:00 to 18:00."
"Then 10:00 or earlier."
"10:00 it is."
"Peachy"