In this paper, we study the differential power operation on ideals. We begin with a focus on monomial ideals in characteristic 0 and find a class of ideals whose differential powers are eventually principal. We also study the containment problem between ordinary and differential powers of ideals, in analogy to earlier work comparing ordinary and symbolic powers of ideals. We further define a possible closure operation on ideals, called the differential closure, in analogy with integral closure and tight closure. We show that this closure operation agrees with taking the radical of an ideal if and only if the ambient ring is a simple $D$-module.