HDB flats are subsidized housing built and owned by the government. Private apartments are usually called
condominiums (condo's) here and have considerably more facilities provided (rentals are double minimum upwards to the sky's the limit.).
condo's may be freehold, 99 year or 999 year leases. All HDB flats are 99 year only. You don't have swimming pools or other facilities in HDB estates. Additionally, if you drive, you have to buy season parking from the URA (Gov't).
condos have pools, gyms, private parking, BBQ's, security and so forth. Around 94% of Singaporeans live in HDB estates. An HDB Flat of 1300 sq/ft will cost around 300K +/- while a condo may be from $400 PSF up to the latest of over $3000 PSF (new record here). You can rent a 1300 sqft HDB for 12-1500 while the same sized condo will be 2x or more higher and currently increasing at around 10-15% months in the prime districts.