You have to understand the meanings of 理 and 了. 理觧 means to solve with logic ie deep understanding. Taking a math example, getting a derivative. Usually we just apply the formula. If you can 理 觧 the process, you can derive the formula!

理解=understand 了解=know

Not the same. 理解，understand ,get it,fathom 了解，know about ,learn(about)

Thanks Icy Zhu, Bonnie Tan, Anna Chow and Jeff for all of your answers

lijie for example, you can understand why he/she gets mad when other people saying something bad to him/her. you can say i lijie him/her,because he/she has his/her own reasons. or you can understand what the question's meaning so you can give the answer to it. you can say I lijie the meaning of the question. liaojie for example,you know the person very well , you will know what he/she will do in some situations. you can say i liaojie he/she. or another meanings of liaojie just like some people explain to you something，when you got it，you will say i see（i liaojie ）， exactly like i see or It explains the matter.