Since, as you write, the drivers are not available from the Ubuntu repository (yet), you could install the drivers directly from the Nvidia web site. Apparently, it explicitly supports GT 630M.
I have no idea why it works in the live system. Of course, you can find out what driver and in which version is used by the X. You will find this information in the Xorg log files in
/var/log. For example, one of my systems shows the following information:
(II) LoadModule: "intel"
(II) Loading /usr/lib/xorg/modules/drivers/intel_drv.so
(II) Module intel: vendor="X.Org Foundation"
compiled for 1.7.6, module version = 2.9.1
Module class: X.Org Video Driver
ABI class: X.Org Video Driver, version 6.0
(II) intel: Driver for Intel Integrated Graphics Chipsets: i810,
i810-dc100, i810e, i815, i830M, 845G, 852GM/855GM, 865G, 915G,
E7221 (i915), 915GM, 945G, 945GM, 945GME, Pineview GM, Pineview G,
965G, G35, 965Q, 946GZ, 965GM, 965GME/GLE, G33, Q35, Q33, GM45,
4 Series, G45/G43, Q45/Q43, G41, B43, Clarkdale, Arrandale
The Xorg drivers are in packages with names like xserver-xorg-video-name, so one can find names and versions of the installed drivers as follows:
dpkg -l | grep xserver-xorg-video
That way you can compare the system from the Live CD with the installed system.