what is the difference between the convolution and differentiation over image in image processing?
In this context, convolution refers to the operation of LTI filtering performed on the image by the filter impulse response. Filtering can have many purposes such as blurring, sharpening, noise reduction etc. 

Certain applications requires that you compute an approximation to the derivative of the image data. This can be accomplished by filering the image with a specific filter kernel (impulse response) that's some sort of a high pass characteristics. 

Therefore, computing the (approximate) derivative of an image can be accomplished by LTI filtering with a highpass impulse response, which refers to a convolution operation. This is the connection in between.