A playback monitor is usually a higher quality than a standard TV. It will have professional inputs such as BNC connectors and a better picture tube or LCD screen. It will be able to play back (usually) composite, component and possibly SDI digital video. It has controls not available on a TV like underscan, cross pulse, blue only and others.
A TV will have a demodulator with RF inputs. It can have all of the above playback abilities but will usually had RCA-type connectors or HDMI for digital inputs. It won't have any of the extra controls the monitor has.
A computer monitor will have (usually) RGBS inputs and a D connector type jack on the monitor. It could also have digital DVI inputs or a combination of the two.
Use the playback monitor for quality control, the computer monitor to see what you are editing on your computer and the TV to watch it on the air once you finish your spot!
Life moves pretty fast. If you don't stop and look around once in a while, you could miss it.--Ferris Bueller