dy/dx and y' are different notations for the same thing. The previous form is more explicit and tells us both the function name and the variable with which we are taking a derivative. y' is used as shorthand and implies the normal variable being used for the problem.
For example if you have: x+y = 2y then the derivative of this equation with respect to x would be 1+dy/dx = 2dy/dx OR 1+y'=2y'. I like using the second way because it looks better and takes less time to write.