We introduce Symbolic Alternating Finite Automata (s-AFA) as an expressive, succinct, and decidable model for describing sets of finite sequences over arbitrary alphabets. Boolean operations over s-AFAs have linear complexity, which is in sharp contrast with the quadratic cost of intersection and union for non-alternating symbolic automata. Due to this succinctness, emptiness and equivalence checking are PSpace-hard. We introduce an algorithm for checking the equivalence of two s-AFAs based on bisimulation up to congruence. This algorithm allows us to exploit the power of SAT and SMT solvers to efficiently search the state space of the s-AFAs. We evaluate our decision procedure on two verification and security applications: 1) checking satisfiability of linear temporal logic formulas over finite traces, and 2) checking equivalence of Boolean combinations of regular expressions. Our experiments show that our technique often outperforms existing techniques and it can be beneficial in both such applications.