How Subways and Trains Differ
While the most obvious difference between subways and trains is where they run, with one underground and one aboveground, there are other general differences. In truth, a subway could easily run as an aboveground train, as there's no needed functional difference. However, there tend to be differences between them to make each more suited to their role.
Subways are generally meant for passenger travel only, with a shape and style not designed to carry freight. Especially with the move towards standard-size containers for freight (a size that would be uncomfortably large in subway tunnels, as well as non-aerodynamic), subways are entirely unsuitable for freight transport. Focusing on passengers allows the subways to make their trains highly aerodynamic in design, the nonstandard but useful form allowing many subway lines to reach much higher speeds than an average rail line. Subways are designed with passengers and speed in mind first, with little concern for storage room for freight.
Similarly, as far as speed goes, more subways rely on electrical and magnetic systems than trains. The 'third rail' is more associated with subways because subways tend to use the electrical system that calls for one more often. While maglev surface trains exist, they are rarer than the sort of high-speed electrical trains in subways.
Of course, range is another matter. Due to the difficulty of making tunnels beyond a certain length, subways tend to be designed for much shorter trips overall. Trains, meanwhile, are made for longer journeys. One will rarely see a dining car or a snack car on a subway, as there's no need for them over the short, fast journeys a subway makes. (Observation cars are, rather obviously, completely out of the picture.)
Because of these shorter distances, subways tend to make more stops, and often use systems that therefore allow for relatively quick braking. Trains tend to slowly reduce speed before reaching a stop, while if the same time was allowed for subways they would rarely if ever actually be at full speed. Subways therefore have a much shorter braking period involved.
However, while it was mentioned before, it's worth repeating that subway trains are generally capable of running aboveground as well - in many places, like New York and California's Bay Area, there are train lines that run both above and below ground. A standard train might have more difficulty going underground, purely because of size, but it would be possible with the right cars. All the differences that tend to exist are sheer matters of practicality.