The Xbox 360 is capable of 1080i resolutions, but only using the component cables. If you use S-Video or Composite cables, you're limited to 480p.
Depending on the VGA adapter you get, it may do upscaling to higher resolutions, it it may just convert the signal and pass it along. The typical 17" LCD resolution is 1024x768 IIRC, which is higher than the standard 480i/p NTSC resolution of 720x480, so you will see black borders around the image from the game that makes up the remaining space if the adapter does no upscaling. If it does do upscaling, you will be limited to specific resolutions, the next one up being 720p at a resolution of 1280x720. This will likely exceed the boundaries of a 17" monitor and so part of the game display will be cut off.
The enjoyment factor will come down almost exclusively to the response rate of your monitor. Anything over 16ms will likely be too slow for gaming, and will result in very noticeable ghosting effects. Anything under 16ms will be fine. The other issue is the contrast ratio, and how well the monitor does chroma sampling. If the contrast ratio is low, everything will seem dark, and some dark areas of games may be completely black. Anything below an 800:1 contrast ratio I wouldn't recommend trying.
The image will not look sufficiently different from what you see on your TV. It might even be smaller, since you're losing about 9" of screen space. However, sometimes smaller makes things look smoother and less jagged.
Personally, I don't know that it's really going to be worth the cost of the adapter for you. The 9" of lost screen real estate are really going to hurt. If it were a 24" LCD, it might be worth it, but I don't really think a 17" is going to cut it. I only use my monitor as a TV for games because of space restrictions where I am. My longer term goal is to get myself a nice HDTV set and hook up all my consoles to that.
I was thinking about connecting my xbox 360 to my 17" lcd pc monitor. I have heard that vga adapters can make this possible. I was just wondering since my monitor can reach high resolutions. Does this means that the games will be high def on the monitor, or just regular looking. Also how enjoyable is this. I'm used to playing games on my 27 inch sony.