What is the difference between radical ideal and the radical of an ideal?

It depends on your definition of "radical ideal". If it is:

An ideal $J$ is radical if and only if there is an ideal $I$ such that $J=sqrt I$.

Then, yes. If it is the more common version:

An ideal $J$ is radical if and only if $sqrt J=J$.

Then, no: you still have to prove that $sqrt{sqrt{I}}=sqrt I$. The two definitions are equivalent, pretty much because of the identity $sqrt{sqrt{I}}=sqrt I$.