Open up a shell and launch Xonotic from there. Post the output of Xonotic inside the terminal window.
>My system is an AMD Athlon X3 and ATI integrated HD 4250
Hm, your graphic adapter might cause issues here. Xonotic is pretty demanding, although a GeForce 6600 was capable of running it.
Unfortunately Q3 is not really demanding and as such it does not mean anything if you can run Q3 but aim for running Xonotic.
Even my old GeForce 4 Ti 4400 can run Q3A, OpenArena and such with the free Noveau drivers