Rāmāyaṇa 2.2.40 notes—vyasaneṣu manuṣyāṇāṃ bhṛśaṃ bhavati duḥkhitaḥ: “[Rāmacandra] is very sorry among people in calamity.” (Rāmāyaṇa 2.2.40)

Even though this was Lord Rāma’s nature, and He noticed His mother’s calamity, He apprehended that this would become an obstacle in carrying out His dharma of obeying the [dhārmika] instruction of His father and steadied Himself. [This is evident from the text.] Therefore, it is inferred that after obtaining His mother’s  permission to depart to the forest, He cried to express His love for His mother. This was to prevent her from succumbing to sorrow.