Reading through Delilah's wiki page, it seems like Euhorn Kaldwin just didn't like Delilah very much.
[She] spent most of her days ... hoping that one day, her father would show her the same attention he showed her legitimate younger sister.
You could take this to imply that her illegitimacy was his reason for preferring Jessamine. However, it appears he didn't think much of her mother either:
Delilah was whipped repeatedly and thrown out of Dunwall Tower alongside her mother.
It appears to have been the Emperor's decision to kick them out of the court, in the same way Jessamine would have decided to let Emily stay. I don't need to justify that Jessamine always loves Corvo, and by extension, their daughter. She never married anyway, so had no legitimate heir to replace Emily with. (She would have ignored or forbid any gossip about where Emily came from. She never lets her advisers tell her what to do — that's why they assassinate her.)
The mechanics of inheritance are up for debate; Delilah claims she is the rightful Empress even through she was illegitimate, whereas Euhorn clearly thought otherwise. Perhaps the crown passes to illegitimate children if and only if there aren't any legitimate ones. Otherwise, IRL, some monarchies let their reigning sovereign 'legitimate' an illegitimate child, i.e. they inherit because the current monarch said they inherit. Jessamine might have decreed that Emily got the throne.
While no one was certain Emily was Corvo's daughter, Treavor Pendleton certainly guessed. He taunts Corvo about it when he's injured on Kingsparrow Island:
"Everyone knows you were screwing the Empress."
Pendleton has no fear of being executed for treason at that point. His wording suggests other nobles assumed the same thing, just kept their mouths shut. Euhorn Kaldwin's page doesn't tell us if anyone knew Delilah's parentage, only that her mother worked as a maid in the tower kitchens. It's conceivable that he denied Delilah was his daughter, and people chose to believe him because he was the emperor.
Euhorn Kaldwin didn't like Delilah as much as Jessamine. Fortunately for him and Jessamine, Jessamine was 'more legitimate' than her, so he intentionally swept her under the rug/bus. Jessamine did love Emily, her only child. In the absence of a more rightful heir, or perhaps by imperial mandate, Emily inherited the throne from her mother.