The brand of a card will affect the clock speed of the cards processors, the kind of cooling setup it has, sometimes the output options and customer service/quality. They all are based on the same chip and should give performance which is within 10% of each other. Cuda cores aren't actually relevant for gaming, I just used them because it was the one stat nividias website listed for both cards. but I just did a quick bit of googling and 260ms aren't actually 200 series cards, they are modified 9800s and give similar performance. 125MB of memory isn't a huge difference and if the 260 has DDR4/5 memory, it will have an advantage over the 260m which has DDR3. Basically don't swap your 260m into the desktop, the 260 will be the same or better.

The 9600 isn't a integrated graphics card, most likely your friend was using it as a physx ( http://www.nvidia.com/object/physx_faq.html ) card or to run extra monitors. Videocards with M (mobile) at the end of there name are laptop cards which are inferior to there comparable fullsize cards, so i doubt you would see any performance gain if you could use your 260m. (for example a full size 260 has 192 cuda cores, a 260m has 96 That computer is a pretty good deal, you will most likely want to upgrade the videocard soonish but it will give better performance then your laptop, due to the quad core and ddr3 ram.