Search-optimization problems are plentiful in scientific and engineering domains. MaxSAT and answer set programming with weak constraints (ASPWC) are popular frameworks for modeling and solving search problems with optimization criteria. There is a solid understanding on how SAT relates to ASP. Yet, the question on how MaxSAT relates to ASP-WC is not trivial. The answer to this question provides us with the means for cross fertilization between distinct subareas of automated reasoning. In this paper, we propose a weighted abstract modular framework that allows us to (i) capture MaxSAT and ASP-WC and (ii) state the exact link between these distinct paradigms. These findings translate, for instance, into the immediate possibility of utilizing MaxSAT solvers for finding solutions to ASP-WC programs.