Jim LOTFP has been trending towards placing his LOTFP material in an alternate earth and featuring more explicit horror adventures. It does raise the question, beyond system, what's the difference between LOTFP and a period-based Call of Cthulhu game?
The difference in tone is huge.
In Call of Cthulhu (COC), characters have a moral position. COC characters investigate eldritch horrors from a sense of responsibility, thus the frequent "save the world" story lines in Call of Cthulhu games. This lends a tragic undertone to the sanity mechanic - you are putting your character in harm's way and will eventually be driven mad or forced to retire.
In D&D style play, even with the horror theme, the party traditionally consists of roguish sorts motivated by treasure. If they run into the occasional eldritch horror, it's an occupational hazard that turns the game into a desperate fiasco to survive. Certainly not a mandate to do something heroic. This is why the sandbox works so well for the one approach, while the other assumes the characters will see the plot through to a finale.
On a separate note, would you refer to a period-based horror game as "alternate earth", "secret history", "pseudo-historical", or something else? These are the burning questions for which G+ excels, and I'll inquire there as well.