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They are very much alike.
Similarities (without spoilers):
-Both male and female leads of both dramas all have a birth secret.
-Strong females
-Both male leads are forced to hide for a living. Rim is locked away in the palace and Nokdu is forced to lay low on islands.
-Female leads are smart
-Both female leads aren't afraid to bad mouth the King

Differences (without spoilers):
-Lee Rim is smart but he's very dependent on others due to being confined most of his life.
-Nokdu is smart and strong, he is independent
-Dongjoo is strong and so is Haeryung however, I would venture to say Dongjoo is stronger

SPOILERS BELOW:
Similarities:
-Lee Rim and Nokdu are both rightful Crown Princes
-Both Rim and Nokdu don't want the throne, they'd rather have a simple life.
-The Queen's (Queen Dowager in Rim's case) are very loving of their son/grandson

Differences:
-The King (the real one, Rim's father) in Rookie Historian is dead
-The King in Tale of Nokdu is alive and tried to kill Nokdu when he was born

They have passionate male leads that are head over heels and willing to do anything for their woman

pretty similar to the fact that it takes place in a high school and there’s a group of students that come together to solve a mystery. In white christmas, a psychologist comes into the high school in this drama, the teacher does but in both cases the students are manipulated into doing things

both have Kim Kang Woo, and both have unique never seen before in kdramas/kmovies plots. one is a futuristic alien story and the other is a supernatural urban fantasy kind of a story

both plots are about people that "accidentally" become detectives. they are both comedy mystery genres and overall a fun watch

I'll just connect both 1st seasons together. they are both fun mystery variety shows that lets the viewers guess along side the participating members.

both dramas are female centered, and their comedic sense is slightly similar. overall both are a fun watch

I love recommending/connecting dramas that have my sense of comedy? they are not over the top but they are ridicules and funny. kim bomi and Jung Bok Dong are kinda similar too xD

both male leads are the sweet dorky kind. while the female leads are different from the usual representation of a female character in dramaworld.
both couple also are a "friends to lovers" kind of a ship~

both kinda break "gender roles"? SWDBS has a freakishly strong female lead that saved the day and caches the male lead when he falls. while TTON has a male lead that disguises as a woman and he looks soft and pretty.

recommendations are not just about similarities. if you watched mr sunshine 1st, then her private life could be like healing medicine?, and if it's the other way around then please beware, mr sunshine is not a romcom ;p
but both male leads have that serious yet dorky side? and both female leads are strong and independent women~

both are romcom office dramas. both female leads are independent and smart though granted, no one can beat mr lion :)

Both have many political schemes. smart protagonists and you always crave for more eps!

In Nirvana in Fire, the protagonist succeeds in every situation while in Royal Nirvana, the main character fails a lot and you suffer more when you watch it.

Both are excellent in their own ways :)

Both protagonists are very smart, come back with different names and plot a revenge against all people who crossed their ways.

if you liked one of these then you'll surly enjoy the other too. they are not really that similar in plot or characters, but as someone who always looks for dramas with great female characters, I love connecting these dramas ;)