슬프다 means being sad from seeing death or other misfortune. It may or may not involve oneself.
서럽다 means being sad with a bitterness caused by sufferings or unjust treatments.
It describes an outburst of emotion owing to long suppressed anger and bitterness about being wronged or persecuted.
So 서럽다 is connected with 원통하다 (feel embittered), 분하다(feel mortified), or 억울하다 (feel wronged), which are rather frequently encountered words in Korean compared with English.
In comparison, 슾프다 is a calmer and more resigned emotion than 서럽다.
- 나는 친구의 죽음이 너무나 슬펐다. (Sad, more or less as a bystander)
- 3년만에 감옥에서 나온 그는 서럽게 울었다. (Sad and embittered about one's own situation)