Tightslot - nicely summed up.
1000 feet is the minimum vertical separation in many circumstances, about 330 metres, or the equivalent of about 4 - 5 wingspans of something like a 777/747/380 - so it's not that far, but it's regulated and safe. And yes, in the right circumstances you may well see the pilots faces' - what a terrible thought
bpaj - FWIW in some parts of the world in some circumstances,( usually at low altitude) , the approved vertical separation is reduced to 500 feet - now that does look close....and certainly gets the aforementioned TCAS grumbling....
If you want real fun go to States and you'll see heavies doing almost formation approaches to parallel runways (e.g. SFO, IAD), the 5 miles/1000' figure you quote is not a blanket rule.