Both are revolving around a group of police officers that work together to solve crime cases. In both dramas those cases have a special link to the officers. Both dramas deepen the cases by also focusing on the suspects and the victims. Both thematize cases that grip the viewer.
Cinematographic-wise and also music-wise both are also working with the same elements.

Both dramas cover crime cases that dig into the cases by giving each suspect a background story, dramatized with flashbacks as to why they did what they did. Both dramas have gripping cases and stories that make the viewer curious what will happen next.

Both dramas have a case that is connected to characters that share a tragic past. In both dramas the focus lays on the detectives who investigate the case and the people who are suspects. Both give of a similar vibe regarding the emotional aspect of the cases and the tragic background story.
Both dramas question the topic of "is the law-system always right?"

In both dramas a woman died in a building. The other tenants of that building are connected to the victim. Throughout both dramas one is left wondering who is the killer and what has happened. Topics such as class-differences, unfaithfulness, lies and manipulation are in both dramas a topic.

Both dramas deal around a group of children that get involved in a murder case. In both dramas especially one kid becomes a suspect and faces doubt by his surrounding. Both are very similar in terms of the children and cases that they face

Both c-dramas are done by the same director and also written by the same screenwriter. Both are making one stay glued to the screen, wondering about the truth behind what has happened. Both are absolutely gripping thrillers that are unique and very well done

A mysterious person that takes up the murderous requests of people. In both drama these people accept a bargain, while in Jigoku Shoujo its their soul that ends up in hell, in Imposibility Defense its death if their request is not pure.

Both deal about a male duo that works together but where the older one is neither good or bad and where the younger one is suspicious of them. Both dramas show that the characters have both had a dramatic past that seems connected to each other. The only big difference is that Watcher consists of police officers working together with a lawyer, whereas The Devil Judge is a pure lawyer drama.

Both dramas have a couple that is meeting due to unusual circumstance again but where one of them makes the decision to help the other. By doing so all actors get involved in a case that challenges not only the relationship they have, but also the characters development too.
There are a lot more similarities between them however naming them would spoiler too much.
IF you enjoy one there is a high chance you will enjoy the other drama too.

Both were released nearly at the same time and both are absolutely gripping dramas that will leave you guessing on what happened and who is responsible for it.
Both have absolutely amazing cinematography and background ist, astonishing acting and interesting characters that are so diverse and complex.

The only difference is that while Burning Ice focuses on one specific case with multiple parties involved, Day and Night focuses more on a variety of cases.

Nonetheless there is a high chance that if you like one you will enjoy the other drama.

Both are crime dramas where the leader of the investigation team is a woman. Both women are very smart and skilled to find the culprits of the cases they handle. Both women are challenged by the work and family situation.
Both give off a very similar feeling.

Both dramas revolve around a character who gets confronted with the death of a close family member. In both dramas the character seems to be the one behind that murder. Throughout the drama we learn what has happened. Both dramas have characters that have amnesia and slowly throughout the drama get their memories back.
Both are gripping crime dramas that will keep you on your seat.