The incubation period is the time it takes from the infection of a patient with a disease until he develops symptoms. How fast this happens depends on the disease.
According to the references the main difference is that SARS 2003 replicates in the lung/lower airways, while SARS-CoV-2 also replicates in the upper airways and sheds high viral loads there early in the course of the disease.
Since both diseases are transmitted by droplets, it is much easier to have a sufficient viral load in infected cells from the upper airways compared to the lower ones.
References:
1. Severe Acute Respiratory Syndrome Coronavirus Infection of Human Ciliated Airway Epithelia: Role of Ciliated Cells in Viral Spread in the Conducting Airways of the Lungs
2. Virological assessment of hospitalized patients with COVID-2019