I have walked across it on the John Muir Way which is highly recommended. I actually didn't really remember what Hadrian's wall was. We always learnt it was to "keep out the Scots", but in fact it represented the Northernmost border of the Roman empire. I had no idea about the Antonine wall, nor that they got that far north.
The Scots are descended from an Irish Gaelic (Celtic) tribe who migrated from Ireland to Scotland in the 5th Century [0] (when all three of Britain's countries were created, it was a fascinating century).
The Romans were there before then, and left before then. The walls they built were to keep the Picts out (though this gets fuzzy - the line between "Pict" and "Briton" isn't as clear as conventional Victorian history books say).
One of the interesting things that I heard about the walls, and may or may not be true (I'd be interested if anyone has an update) is that the Romans never explored the top of the island, or sailed around it, and just assumed there was a lot more of it going north. If they'd known how close they were to the end, they might have just conquered all of it, which would probably have been less effort than building those two walls.
Just to be clear, "Scots" means current denizens of modern Scotland, who are mostly descended from a mixture of Picts and Gaels. It's not the case that the 5th century "Scoti" wiped out everyone else and they're now the only modern "Scots". The Picts and Gaels eventually united under Kenneth MacAlpin as the Kingdom of Alba... which did not include Strathclyde and the Lothians as Scotland has today.
Also to be clear to anyone else reading, "Britons" is referring to https://en.wikipedia.org/wiki/Celtic_Britons rather than all citizens of modern Britain. These people were predominantly in Strathclyde and Cumbria. I'm not sure how many of them were around in Roman times to be kept out.
As you say, the Romans departure from north Britain predates "Scotland". They fought with many different tribes north of Hadrian's Wall and the Antonine Wall. From Ptolemy, we have some of the names they gave these tribes: Taexali, Vacomagi, Caledonii, ... We don't know if the tribes call themselves that. See https://en.wikipedia.org/wiki/Scotland_during_the_Roman_Empi... -> https://commons.wikimedia.org/wiki/File:Britain.north.people... which is based on a 1467 map, which itself is based on Ptolemy's writing.
