Heavy Rail (rapid transit) is also known as "third rail". That third rail is electric, allowing for higher speeds and rapid acceleration. These types of systems also tend to have shorter headways and higher passenger capacity than other forms of rail. Because of that third rail, they also have to be grade separated.
Commuter rail/regional rail is an electric or diesel propelled urban passenger train service consisting of local short distance travel operating between adjacent cities and towns, or between a central city and adjacent suburbs, using either locomotive hauled or multiple unit railroad passenger cars. Many commuter rail systems only run on weekdays and have longer headways than heavy rail rapid transit systems. Also, unlike heavy rail, commuter rail systems can operate on the same track as freight rail.