your tv has a NATIVE RESOLUTON of 1280x720
Here is what is happening when you output 1080p from your 360 onto your TV:
The game code renders an image that is 1280x720...
the Xbox 360's scaler then stretches that image to 1920x1080 using guess work calculations to fill in the gaps in the new larger size and then converts it to a component video signal
your TV then decodes the component video signal and then squishes 1920x1080 down to 1280x720 throwing out lots of pixel data and changing some pixel data to get the image to fit in the new smaller size...
So what happened
1. you make both your Xbox 360 and TV work harder by performing needless operations, this could potentially introduce lag, make them run warmer, and a number of other things
2. you ended up with a resultant image that isn't true to the image that was originally generated by the game... the TV doesn't know which pixels were created by the game and which ones were added by the scaler so when it changes or throws out pixel data it's changing and throwing out just as much of the original game pixels as it is superfluous scaler added pixels
3. you have a higher probability of data corruption and noise using 1080p over component, there is more data and it has to travel at a higher speed which increases the chances of the image getting muddied up by the analog cables you 're using... The reason there is rare support for 1080p over component is because most Component cables can't handle the speed well enough to transmit the image cleanly.
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Here's what would happen if you used 720p output:
The game code renders an image that is 1280x720...
the Xbox 360 doesn't use the scaler at all because it doesn't need to. the un-altered image is then converted it to a component video signal
your TV then decodes the component video signal and sees that it doesn't need to use it's scaler and sends the un-altered image directly to the pixels on the screen.
I'm not going to tell you which resolution to choose... I should hope the answer is clear