Closing mobility and accessibility gaps between public transit riders and private car users is key to tackling social exclusion and achieving sustainable development goals (SDGs). However, place-based potential accessibility methods do not accurately measure real gaps in the uptake of activity opportunities because people usually have limited activity spaces. This study introduces people-based activity space approaches to measure activity disparities between the two modal groups. To overcome difficulties in obtaining large-scale individual activity data, this study used vehicle plate recognition data and public transit smart card data to anonymously identify activities. Individual activity spaces were characterised by six primary activity features from different dimensions. The analysis confirmed that, relative to transit riders, people who use cars on average accessed more activities within a larger activity space and enjoyed overall higher travel efficiency. A comprehensive indicator was further derived from the primary activity features to quantify activity disparities at the zone level. Zones with the highest risk of social exclusion were observed in the outskirts. In contrast, the city centre and inner suburbs exhibited significant equality of the two transport modes in fulfilling mobility needs for engagement in activities. Activity disparities between the two modalities were determined per area in specific activity dimensions, namely activity extensity, activity diversity, and travel efficiency. Finally, statistical models provided evidence that public transport facilities (especially rail transit) and location factors (distance to the city centre) are essential in determining modality-associated gaps in access to urban activity opportunities. Socioeconomic status and land use diversity also partially contributed to the inequality in specific dimensions of the activity space. This people-centred approach is critical for tackling transport inequality and achieving SDGs while ``leaving no one behind''.