비슷하다(bi-seus-ha-da, biseushada)
Indicates that two things are almost the same or similar.
When to use
Appearance comparison, Personality comparison, Opinion similarity
The nuance
Use 비슷하다 when two subjects share many common points or similarities. They do not have to be identical; it applies to resemblance in shape, personality, ideas, etc.
다르다(da-reu-da, dareuda)
Indicates that two things are not the same and can be distinguished from each other.
When to use
Preference difference, Personality difference, Opinion disagreement
The nuance
Use 다르다 when two subjects are distinguishable or have clear differences in nature, state, thoughts, opinions, etc.