Similar premise...
Lost is tad more mature and way more emotional than one spring night.... the feeling and the emotions they provoke are kind of same though Lost has some scenes where you won't be able to stop your tears.... I would totally recommend watching both of them...

Though both are two different genres…. The feels are the same with similar small town setting and psychological suspense. Hometown is a psychological thriller based on a cult involving mass brainwashing and killing while Monstrous is a horror suspense with an evil spirit that turns humans into monsters by affecting their minds…

Both the dramas depict FL who are misunderstood celebrities and deal with loneliness and negativity from their fans/haters. Both have cute romance. Apart from that both the dramas have different settings. The ML in The Beauty Inside is a rich chaebol suffering from prosopagnosia while in Fanletter, Please ML is a lawyer who has a sick daughter.

Similarities- One of the leads are cursed in both. In TBI the FL while in DWY it’s the ML. The ML in both dramas have similar characters… Being tall handsome and cold which changes when they meet the quirky FL. Overall the feel of both dramas are also similar with sweet romance and quirky comedy in between.

Both dramas involve people with superpowers living on the run or hiding. In both there are organizations who are after the leads.
Apart from that the thriller and gore vibes are kind of similar... although connect is way too much gorier....
Moving has better plot progression and more characters and good character development...
Personally I prefer Moving more...

Both dramas involve time travel and murder mystery.... more over the feel is kind of similar in both along with some bromance...
Apart from that the plot in MB also includes superpowers...

Similar in-
Husband’s debt/Betrayal.
ML approaches FL with secret motives.
FL has difficulty trusting the ML at some point in the story.