An "independent" executor, usually called an administrator, is not really at all different from the executor under a will.  The differences all come from the fact that in one instance the deceased left a will, and the other deceased did not.

The executor and the administrator both have the same job, as personal representative of the estate.  The executor gets the job by being named in the will, the administrator gets the job by applying for it as one of the deceased's next of kin.  Once the bills are all settled, the executor distributes the net estate according to the directions in the will, while the administrator follows the statute that's used when there is no will ("intestate succession").  Otherwise, the powers and duties are identical.