Question: I want to express my great satisfaction of this person. Are there any differences between the two sentences?

> 
> This shows a great sense of responsibility which is **deeply**
>  appreciated.
> 
> 
> This shows a great sense of responsibility which is **dearly**
>  appreciated.
> 
> 
>

Answer: 
Being a possible collocation of *appreciate*, "deeply appreciated" would be more suitable. *Dearly*, on the other hand, will also intensify the meaning of appreciate by adding a sense of "very much".
There's a *deep* difference between the two.

Something *deeply* appreciated is *profoundly* or *thoroughly* approved.

Something *dearly* appreciated is done so with *fondness* or *affection*, which is not the sentiment you want to convey in this case.