It doesn't help that Scilly isn't usually used in isolation. Usually the group is referred to as the Isles of Scilly or the Scilly Isles. None of the islands is actually called Scilly.
They are also sometimes called 'the last piece of England', reflecting both their westerly position and the fact they are a place where some aspects of time have stood still.
The boat that takes tourists there is called 'The Scillonian'.
I imagine more people in the UK know they exist than you perceive. Anyone who still watches the weather updates on the TV or listens to them on the radio instead of getting it from their phone will have heard mention of the...
Though... the more time goes on, the more I expect to hear "oh, they still broadcast the weather on TV? I thought they just did that during the war"... and by war, they're referring to the Gulf war... along with "Walkman/Discman? What's that?"