Gapminder.org has some pretty fascinating data visualizations, and while the map version
isn't in my opinion the most interesting way to view most of the data, it is the one that fits this thread. (I like the bubble charts myself, as you can see three different statistics at once for each country, and animate the whole display over time.) One interesting thing to do with the map is show by life expectancy
and then animate it (press the play button at the bottom left) to see countries pulse with wars and famine, and to see everywhere in the world shrink a bit around 1918.
I noticed someone else already linked to windytv.com, which I haven't gotten a chance to explore much, but my goto global atmospheric and oceanic animated map is still this one
Another example I could use. I can go to my nearest beach and stand on a point where due west there is nothing (but water) between me and Argentina, and due south, nothing between me and Antarctica.
Both of these can be shown to children on a Mercator projection map of the world or a globe. But it would be hard for them to visualise that on the map above. It is too distorted to be usable.
Well sure, as was already acknowledged that map distorts or breaks up the oceans, so obviously it's not the one you should use to point out oceanic facts. Meanwhile one of the few useful things about Mercator is that it does
preserve bearings in every direction.
(Of course, the "straight line" great circle path starting due west from Perth hits Madagascar and then Tanzania. You have to start out at a bearing of 185º to go directly to Buenos Aires from there.)