If a person swore an oath concerning something he believed he did not remember, then remembered it after swearing and it became clear to him that it was contrary to what he had sworn, is this considered a vain oath (halaf laghw) that only requires repentance, or an oath that necessitates expiation (kaffarah)? And what is the difference between the two?

Merely informing contrary to reality is blameworthy lying, and swearing an oath upon it is even worse. The questioner must repent and seek forgiveness. If the questioner intended to make an oath, then it is a "Ghamus" oath, and sincere repentance is obligatory without expiation according to the majority of scholars, in contrast to the Shafi'is. If the oath was uttered unintentionally without purpose, then it is a "laghw" (unintended) oath, and there is no expiation for it, leaving only the act of mere lying.