The tone and scale is quite different, but the overall themes or message certainly have a lot in common. Both worth seeing.

I can't remember The Day After as clearly as I can Threads, but I remember it was definitely worth seeing, though I feel like it had a little more of a film/plot/narrative/entertainment element to it, whereas Threads was just quietly bleak and undignified - a gritty soap-opera story in Sheffield, then everyone gets nuked and you see all your favourite characters as they piss themselves and their hair and teeth fall out from the radiation and then they slowly die from illness and starvation, then you watch a documentary style presentation of the tiny remainder of population, scrabbling in dirt, trying to find a still living plant.

Absolutely watch both - all humans should watch both at some point in their lives - but maybe not on a day where you want to have any fun or talk to anyone, or do much except stare into the distance in silence.