Is there a difference in the way the old Xbox360 reads discs and the new model?

There is no difference in console models. Although, they do use different disc drives in different batches. For example, a fat 360, can have a LG drive, while another fat 360 can have a Samsung drive.

You are having problems with scratched discs, because your drive is not able to read the data. Technically, it should be unplayable in such case. But, your friend is kinda lucky here, that his drive is able to read the games. There is nothing that can be done in this case, but to get the disc replaced. Also, if this happens to new games too, try getting console replaced. I had similar issue with BF3, got console replaced, and new one reads the game fine.