Deletion operations play a fundamental role in the formal language theory, with applications spanning both computational and biological systems. Deletions with specific constraints, in particular, site-directed deletion, are of special interest for applications. In this paper, we examine the computational and structural differences between deletion closure and site-directed deletion closure operations, addressing some open questions. First, we prove that any recursively enumerable language can be obtained as the site-directed deletion closure of a context-free language. We observe several differences between deletion and site-directed deletion applied to context-free languages. Secondly, we establish that regular languages are closed under deletion closure, resolving an open question from [Ito, Kari, Thierrin 1997]; the closure of regular languages under site-directed deletion closure remains an open problem. In contrast, we show that closures of regular languages under deletion systems are computationally universal. Finally, we show that the membership problem for site-directed deletion closure is NP-hard. These findings contribute to a deeper understanding of complexity of iterated deletion operations.