The film was made in the mid-eighties when the possibility of an all out nuclear war was still a common fear, it follows a more traditional narrative approach compared to Thread's documentary style. Its made for TV roots shows in the production, but don't let that put you off as it is an excellent, if grim film.
It takes a while to get going, the build up is slow as you're introduced to the various characters (although they are all based in same region of the US). Besides the setting this is the main difference between the two films, Threads focused on one central character where here the cast is much larger (and there's quite a few known faces making it a fun who's who spotting exercise).
Once the action gets going the change is stark, it's not quite as bleak as Threads but it is still a grim sequence of events. The actual nuclear attack is quite well done in a low budget way, but still effective. Only the immediate aftermath of the attack is shown (where Threads stretches further into the future), there is a comment at the end stating that the film probably understimates the impact of an all-out attack, which does drive the point home.
For its time this was a very shocking film and while we have different fears these days it still illustrates the potential we have as a species to damage ourselves. It shows its age, but this is stilla fine, if grim film to watch.
The Day After is available from Amazon (and is a good watch)