I think Pearlmutter and Siskind‘s work is primarily about using ideas from lambda calculus to clean up AD (automatic differentiation) of numeric functions. Whereas differential lambda calculus is not about numeric functions per se: the aim is to define derivatives for arbitrary algorithms.

So, for example, I do not think that in the Pearlmutter-Siskind system that a function of type Int -> Int would have a meaningful derivative, whereas in the differential lambda calculus it does (this is surprising).

I do not recall the details of the Pearlmutter-Siskind system, but some approaches to programming with AD is based on adding formal infinitesimals. There are no infinitesimals explicitly in differential lambda calculus or linear logic, but in the semantics of the latter that we use, derivatives do arise from the coalgebra (k[x]/x^2)^* which is the manifestation of a first-order infinitesimal in algebraic geometry. So in some sense both subjects proceed by adding formal infinitesimals to an existing programming language, but in differential lambda calculus this is implicit.