Sometimes at work (and at home, when I make coffee there, which isn't often anymore), you'll go to get coffee but there won't be any there. So you put on a new pot.
If you're really impatient, you might do the switchout and hold your mug under the filter basket, then put the pot back. Do you ever wonder if you're getting the same cup of coffee that way as you do when you let the pot fill?
I don't think you are, but I don't know how different it would be. Take as a given that the first bit of water that passes through the filter is a bit different from the next, and by the end (a full pot), it's completely different. So, when you let the pot fill, you're getting a mix of everything, beginning to end. When you get a cup directly from the filter, you're only getting that bit, without the full mix.
I think the first bit might be really weak, then spike up to quite a bit stronger, then trail off to the end of the pot to be really weak again. But yeah, I really have no idea.
This would make an excellent science fair experiment.