Don't put much stock in the terms HD or Full HD they don't have consistent meanings. The terms that are important are 720p or 1080p.
720p is 1280x720
1080p is 1920x1080
The p part really means not "i" were "i" stands for interlace, and is not something you want.
A monitor/tv is said to support 720p if it's resolution is at least 1280x720, and 1080p if it's resolution is at least 1920x1080. You can just compare the resolutions directly to find out which one looks better.
The basic differences between a monitor and a television are:
1) A television indeed has a tv tuner
2) Televisions generally have image processing on them to make bad sources look better
3) Televisions have a bit more lag in them, which you might notice when moving the mouse around
4) Televisions are designed to look good from 6' to 15' while monitors are designed for around 20". Cheap televisions in particular might not look right up close.