'Clutch' is what you do with your hands. It's a movement of grabbing hold of something. 'Cling' can be with your arms or your whole body. The essential meaning of clinging on to something means 'not to let go'. You cling on to something for safety or security, and you don't let go.

I think clutch like to a bag or purse or hold. example "I clutched my purse waiting in line to checkout". I think of cling as you don't want to let him, her or it or idea get away. example " I cling to thge memories of my college days when I was happiest." or " I cling to my friends for support during a1 hard time in my life."

Thanks a lot ,got it;)