This paper presents experimental comparisons between the declarative encodings of various computationally hard problems in Answer Set Programming (ASP) and Constraint Logic Programming over Finite Domains (CLP(FD)). The objective is to investigate how solvers in the two domains respond to different problems, highlighting the strengths and weaknesses of their implementations, and suggesting criteria for choosing one approach over the other. Ultimately, the work in this paper is expected to lay the foundations for a transfer of technology between the two domains, for example by suggesting ways to use CLP(FD) in the execution of ASP.