is a full install better than using a live dvd

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Yes! A live DVD is good for testing how it works in your computer and for installing, but you caot save anything, no updates of the system and no data unless you save them in another drive (which is not very convenient). You can also consider creating a USB boot drive and use it similar to a live DVD or as a persistent live drive. And a USB drive can be re-used many times.

Ubuntu and the Ubuntu flavours (Kubuntu, ...) and intended to be installed. See also the following link

Try Ubuntu (Kubuntu, Lubuntu, Xubuntu, ...) before installing it

-o-

But if you use Ubuntu MATE only in order to rescue another system - it is a good idea to use a live DVD. But you must save the files somewhere else

- in an[other] exteal USB drive (hard disk drive or SSD)
- in a USB pendrive (but it is usually smaller and less reliable because of wear of the memory cells).

Everything that is stored in 'the system itself' is actually stored in the volative RAM, which will be gone after shutdown or reboot.