Save the Last Dance for Me (2004)

They were filmed in the same era's I guess you could say, they give the same feel when you're watching it - very calm but nice and romantic at the same time...

both have the same story rich male lead got amnesia and fall for female lead things going to complicated when male lead get back his memories

Eun Soo is a pure but unsophisticated woman who runs a small resort inn with her father. Eun Soo is just an ordinary girl, but her life changes when she meets a man who's completely lost his memory. She takes him in and cares for him, love blossoms, and soon wedding bells can be heard.
But on the night of their wedding ceremony, Eun Soo's father passes away. And soon after, Chang Ho disappears too! But Chang Ho had his reasons for disappearing: his real name is Kang Hyun Woo, and he's heir to a large conglomerate. Even more, someone is plotting to kill him.