This is my view of the connotative meaning - Sadness: the most general "sad" emotion of these bunch Grief: sadness usually felt after the loss of something, such as a death. Sorrow: I see sorrow as in between sadness and grief. It's not as powerful as grief but holds more emotion than sadness. You may feel sorrow after forgetting something, hurting someone's feelings, or failing. Distress: This is tied to a situation that is ongoing and is likely in combination with anxiety or possibly pain. If you are running late, in possible danger, or under extreme stress you may feel distress. Anguish: Somewhat similar to distress, anguish is also related to pain except this time more intensely. It's usually associated with severe mental or physical pain/suffering. If you've just been seriously injured that following emotion could be described as anguish.

They are all synonyms. Grief, sorrow and anguish are usually used for funerals and deaths to express the extreme sadness and feeling of loss. Distress is not only used for sadness over death but also for upsetting situations.