Basically both viruses use the ACE2 receptor to enter cells but, while SARS quickly started infecting lung cells and thus patients had symptoms before they were infectous, covid's virus "learned" to reproduce in the upper respiratory tracts before going to the lungs so a lot of people became infectous before they developed symptoms. Source:
"There is a close genetic relationship between SARS coronavirus (SARS-CoV) and the causative agent of COVID-19, SARS-CoV-2. The predominant expression of ACE2 in the lower respiratory tract is believed to have determined the natural history of SARS as an infection of the lower respiratory tract." [...] "Here we report a detailed virological analysis of nine cases of COVID-19 that provides proof of active virus replication in tissues of the upper respiratory tract. Pharyngeal virus shedding was very high during the first week of symptoms [...]. Infectious virus was readily isolated from samples derived from the throat or lung, but not from stool samples—in spite of high concentrations of virus RNA. Active replication in the throat was confirmed by the presence of viral replicative RNA intermediates in the throat samples. We consistently detected sequence-distinct virus populations in throat and lung samples from one patient, proving independent replication. The shedding of viral RNA from sputum outlasted the end of symptoms."
— Link to full paper