Both examples use the subjunctive mood case, which we use to express wishes, polite requests, and hypothetical situations. The sentence using "would have" indicates that the event occurred in the past and in a similar situation, the speaker would remember the information in question. The sentence using "would remember" only indicates a hypothetical event in the future.

"Would have ____" is used to indicate that it definitely didn't happen. You're imagining an impossible "what if..." situation. "Would" is used for unlikely or unsure possibilities. There's still a chance this could happen.