Both drama deal with a time travel romance with many medical elements. Both fun and sad. Though where live up to your name is more lighthearted, faith is more melodramatic.

Both have:
-Time travel
-romance
-court power battles
-secrets
-lighthearted plot yet with painful moments

-Fun and funny time travel drama with a twist. (Although due to censorship joy of life's time travel was hidden)
-seemingly lighthearted and then the pain attacks you at random points
-secrets and plots
-interesting side character plotlines
-joy of life had more adventure than romance.

Time travel for the win. Amazing and fun main lead, secrets, plots etc. Of course nobunaga concerto focuses more on the politics while mr queen on the romance. Still they give off the same feeling.

they are both cute, fun and at times painful isekai/time travel drama with really good chemistry between the main couple

They give off the same atmosphere
Wintery themes, slow burning romance
Of course A Piece of Your Mind is more calm and toned down.

Both drama have:
-a career woman with a soft/sweet side
-a young talented Main Lead
-Societal Expectations
-Female Friendship
-Strong 2nd Lead Presence
- the topic: what truly makes you happy

Both drama deal with age difference in relationships (older woman-younger man) and the topic of whether one should conform to society's expectations or not. Strong 2nd lead presence too.

•Palace drama dealing with its occupants daily lives throughout many years/decades.
•A smart, magnificent female lead who fell in love with a distrustful emperor that frankly speaking didn't really deserve her affection
•Though they both have a very realistic feel to them, Serenade of Peaceful Joy is indeed more toned-down and peaceful than Ruyi's Royal Love in the Palace.

The main characters are working as labor inspectors and the drama in general revolves around their profession through many cases the main leads encounter. Buuuut Special Inspector Jo deals with many other themes too while Dandarin Rodo Kijun Kantokukan is solely focusing on the work aspect

Firstly they are both fantasy drama. Both deal with prophecies, secrets and polital intrigues. And they both are one story cut in parts.

Both have
-a visually impaired heiress as the main lead
-who the male lead approaches for an other reason than the one they present, hiding their true identity in order to achieve a certain goal.
+the theme of "forbidden love"in both drama.