"Compare" is the word generally used. "Liken" is very old fashioned - it is very rarely used today. They are very similar in meaning, though.

liken is when you are comparing something because they are similar. lemon tastes a bit like a lime the texture of a lemon has a likening to a lime the son has a likening to his father I think you normally dont talk about quality when you use 'liken' but usually you do talk about quality when you use compare. this restaurant has a much better price compared to the other one