Contract-based design is emerging as a unifying compositional paradigm for the specification, design and verification of large-scale complex systems. Yet, different contract frameworks are currently available, without a clear understanding of the relations between them. In this paper, we investigate the relation between interface theories (specifically, relational interfaces) and assume-guarantee (A/G) contracts, revealing some of the subtleties involved. We show that the natural transformation of interfaces to A/G contracts represented by linear temporal logic (LTL) formulas preserves refinement, but does not generally preserve serial composition, and we present an assumption-projection operator to remedy the latter issue. We also discuss the properties of our transformation with respect to conjunction. Finally, we provide illustrative examples that shed light on the effectiveness of both frameworks for requirement formalization, early detection of integration errors, and principled use of abstraction-refinement.