The differences between UNR and Desktop versions are, as you said, just differences on desktop's look.
The base system is ubuntu in both cases, the default installed packages are the same too (just for the record, gparted is available on the live version booted from USB, it's then removed when you install it on HD. It can be easily reinstalled through the package manager).
I still have only few personal files in the UNR so should I just install Desktop over it? Because I made separate partition for /home if I understood correctly even the few files I have can remain there. In the new install I point /, swap and /home to the correct partitions but I do not format /home. Is that right?
Yes, it's perfectly right.
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