'Grieve' is a verb which means to feel and/or show sadness. 'Sadly' is an adverb which means to do something in a sad way. He move sadly because he was grieving for the loss of his friend. She is grieving because her cat died. or She is grieving the loss of her cat. She behaves sadly because of her cat. Sadly, we don't have any drinks left = It is too bad that we don't have any drinks left.

"Sadly" is an adverb, related to "sad", of course. It means sorrowful. People can feel sad for many different reasons. Some suffer from depression. It's a general word. "Grieve", on the other hand, means to feel sad for a specific reason, often due to a loss. "I am sad today." "They grieve for their departed grandmother".