No, they don't mean the same thing.
When you say, "I understand", it means that you have grasped the meaning of a situation, or of a word, or that you sympathise.
When you say, "It makes sense to me", it means, "In my judgment, it is intelligible, or justifiable, or practicable."
The two are not the same in meaning. The perspectives are different. The underlying tone is different. Do not mix up the two.