A group is locally finite if every finite subset generates a finite subgroup. A group of linear transformations is finitary if each element minus the identity is an endomorphism of finite rank. The classification and structure theory for locally finite simple groups splits naturally into two cases—those groups that can be faithfully represented as groups of finitary linear transformations and those groups that are not finitary linear. This paper completes the finitary case. We classify up to isomorphism those infinite, locally finite, simple groups that are finitary linear but not linear.