The usual distinction today is rather subtle; a game will be a "computer game" if it is played on a personal computer (such as Apple Mac, PC, Amiga or Commodore 64), but a "video game" if it is played on a computer that is specialized for game playing (such as video game consoles). Because of hardware differences, versions of the same game played on a computer will typically feature a wider assortment of direct controls exploiting the full computer keyboard, while video games tend to use more layers of sub-menus, or motion sequences (up-up-down-left, etc) via the game controller.
The most important distinction between computer and video games arises from the fact that computers tend to have high resolution monitors, optimized for viewing at close range by one person, while home video game consoles use a much lower-resolution commercial television as their output device, optimized for viewing at a greater distance. As a result, most computer games are intended for single-player or networked multi-player play, while many video games are designed for local multi-player play, with all players viewing the same TV set.
Formerly, video games tended to need and use less computing power than computer games, but with the increasing power of video game hardware, that distinction is nearly erased, and many games are now produced for both computers and video game systems. Video game manufacturers usually exercise tight control over the games that are made available on their systems, so unusual or special-interest games are more likely to only ever appear as games on general-purpose computers.