Many games on both systems are programmed to run at a non-standard (720p or 1080p) resolution in order to make the most of the respective systems' graphical capabilities (e.g. running at a lower video resolution to leave enough for textures or headroom to apply a requisite amount of AA).
The difference then is how the different systems handle scaling. On all but the earliest, launch SKU Xbox 360's, they use a hardware scaler. A dashboard update then allowed for effectively every game to be scaled up to 1080p if they didn't already run at at resolution. Pretty much, if your 360 has an HDMI output port, it has the requisite hardware scaler chip on it.
The PS3 is different, in that it depends on software to scale its games. This is why so many games that are able to advertise and play on 1080p... don't, on the PS3, and I have a feeling it's also why 3D games on PS3 are arbitrarily limited to a maximum of 720p, regardless of the capability of its connected TV. It also explains why some games arbitrarily limit or prevent you from playing them in 1080p on ps3 without an HDMI cable, while every game can play in 1080p on the 360 with even Component Video (YPbPr) cables. -- Because, f*ck Sony