DNA G-quadruplex structures are emerging cancer-specific targets for chemotherapeutics. Ligands that bind to and stabilize DNA G-quadruplexes have the potential to be anti-cancer drugs. Lack of binding selectivity to DNA G-quadruplex over DNA duplex remains a major challenge when attempting to develop G-quadruplex ligands into successful anti-cancer drugs. Thorough understanding of the binding nature of existing non-selective ligands that bind to both DNA quadruplex and DNA duplex will help to address this challenge. Daunomycin and doxorubicin, two commonly used anticancer drugs, are examples of non-selective DNA ligands. In this study, we extended our early all-atom binding simulation studies between doxorubicin and a DNA duplex (d(CGATCG)2) to probe the binding between daunomycin and a parallel DNA quadruplex (d(TGGGGT)4) and DNA duplex. In addition to the end stacking mode, which mimics the mode in the crystal structure, a pure groove binding mode was observed in our free binding simulations. The dynamic and energetic properties of these two binding modes are thoroughly examined, and a detailed comparison is made between DNA quadruplex binding modes and DNA duplex binding modes. Implications on the design of more selective DNA quadruplex ligands are also discussed.