They both deal about possession and evil spirits and the fight against them. They both have the exact same feeling and same degree of brutality.
They both have characters that have special powers to deal with them.

Both dramas deal about strong females that take revenge on the men that wronged them. Both are very similar in terms of feeling, characters and themes.

In both dramas the male lead is played by Joo Sang Wook. Both deal about a female lead that learned form a disappointment that changed her life and works now together with the male lead to either get revenge, as it is the case in Birth of a Beauty, or to help the male lead, like it is the case in Touch. Both dramas have a very similar feeling and a similar relationship between the leads.

Both main female leads have a special ability that makes it possible for them to help an investigation squad. Both dramas deal about murder cases.

Basically the same main lead with a different skill. Instead of being able to memorize everything, she can see sents. Both deal about murder cases, both have a character who is, because of what happened to them, needing the the person with the special ability.

Both dramas deal about a married couple that has to work out the consequences of betrayal. In both dramas the couples work/are from the same business. Both dramas have strong female leads. Generally the theme infidelity is well developed in both dramas.

Both dramas deal about strong independent, cold women, that have one goal and kind of play a double edged sword. At the one hand they can ruin their company, at the other hand they can bring it to success. Both women are accompanied by males that try to help them with achieving their goals.

Both dramas deal about a group of people that tries to fight for koreas sovereignty. Both are quite powerful when it comes to father land, fighting for what one believes to be right and international relations.
Both dramas have well thought through stories with people that jump between the sides. Also both have quite interesting spies.

Both are crime/investigation/Detective dramas, and both are so very complex and twisted that they leave one hanging at the edge of the seat. Absolute amazement.
There are a few more similarities however due to potential spoiler reveal I cannot name them. There is however a high chance that if one liked the one drama, the other one will appeal as well.

Both dramas began airing at nearly the same time and both dramas have quite a lot of similarities. For one we have characters that lost family members and we have cases and people that have something to do with that. We have twisted characters that one, especially in watcher, can't declare as good or evil. Both dramas have very strong female leads . Also, both dramas leave you hanging at the edge of your seat, waiting anxiously what will happen next.

Both deal about school bullying and crimes. In both dramas there is a declared teenager responsible for the crime, however both have more to the story than at first appears to be the case. Both dramas deal about adults that come into that school and with the help of the teachers learn what the school is truly made out of.
Both are quite similar in many aspects.

Both dramas deal school bullying and victims that have more to their story then one sees at first. Class of lies however is having a prosecutor as the main protagonist that tries to undercover what goes on, whereas Beautiful World deals about the victims family and them discovering what has happened.
Both are very similar, so if one enjoyed one, there is a high chance that the other drama will be enjoyed as well.