An outbreak by a new severe acute respiratory syndrome betacoronavirus (SARS-CoV-2) has spread CoronaVirus Disease 2019 (COVID-19) all over the world. Immediately, following studies have confirmed the human Angiotensin-Converting Enzyme 2 (ACE2) as a cellular receptor of viral Spike-Protein (Sp) that mediates the CoV-2 invasion into the pulmonary host cells. Here, we compared the molecular interactions of the viral Sp from previous SARS-CoV-1 of 2002 and SARS-CoV-2 with the host ACE2 protein by in silico analysis of the available experimental structures of Sp-ACE2 complexes. The K417 amino acid residue, located in the region of Sp Receptor-Binding Domain (RBD) of the new coronavirus SARS-CoV-2, showed to have a key role for the binding to the ACE2 N-terminal region. The R426 residue of SARS-CoV-1 Sp-RBD also plays a key role, although by interacting with the central region of the ACE2 sequence. Therefore, our study evidenced peculiarities in the interactions of the two Sp-ACE2 complexes. Our outcomes were consistent with previously reported mutagenesis studies on SARS-CoV-1 and support the idea that a new and different RBD was acquired by SARS-CoV-2. These results have interesting implications and suggest further investigations.