The most important distinction between these two different systems of wastewater removal, is that one is for removing human-caused wastewater and the other for removing natural or environmental wastewater.
In early examples of civil engineering, rainwater and wastewater were removed by the same process, an open channel that ran towards the nearest river or sea.
Later evolutions of wastewater management separated the two into drains and sewers, and buried them.
Modern sewers are designed to quickly remove wastewater from toilets, sinks and other grey water sources through a sewer network to a major trunk sewer -- the main artery that hopefully directs the wastewater on to a treatment plant for processing or recycling. Cheaper but unfortunately less adequate is the old-schoolmethod of dumping the untreated waste into the ocean.
Stormwater drains siphon away urban runoff or floodwater from roads and streets. They tend to be found much closer to the surface than sewers, and follow old water-courses. Comparing two street directories, and the disappearance of a stream over time, or one that ends suddenly at a road, can indicate a small watercourse that has been capped. On occasions larger streams have been turned into stormwater drains due to the building of a major road over the top of them.
Where drains use the natural gradient of the land to keep their contents moving, modern sewerage has to be "pumped" so as to keep pressure in the pipes and stop the wastewater from backlogging out of toilets and sinks. It's interesting to note the emergency measures that have been put in place to manage the sewerage in case of the loss of pump power.
One un-environmentally friendly method is letting the flow drain into the stormwater.
A more aesthetic way is the reservoir.
During my urban exploring days, there were a couple of features that puzzled us - huge, empty washing-machine tub reservoirs built below ground. To give you an indication of their size, they were nicknamed Deathstar and Dreadnaught respectively. No-one at the time knew what their purpose was until a friend of mine with Water Board connections told us that they were sewerage reservoirs (which explained the distant thrumming). In the event of power failure, the sewerage would drain into this rather than back up the good folk of Melbourne's toilets.
Urban Exploration of stormwater drains is a comparatively less hazardous hobby than entering a dedicated sewer. The drain is cleaner and more accessible, and sudden water flows are almost always weather related. Exploring active modern sewers, (as opposed to old "dual" systems such as those found in Paris and London -- most which are no longer operational) is highly dangerous and not recommended.