What is the difference between DNA vs RNA Editing in the context of gene therapy?
There are several differences, both mechanistically and functionally.


DNA edits are heritable by the daughter cells and are therefore permanent whereas RNA edits are not.
The type of Cas protein used in the editing process for DNA and RNA are different. 
In the paper that you linked the authors have fused an RNA editing enzyme (ADAR) with dCas13 (a variant that does not cleave the target). So the CRISPR-Cas system directs ADAR to the sites on RNA that need to be edited. Usually when editing the DNA with CRISPR-Cas, people rely on the ability of the complex to cleave the DNA. This double strand break may be then repaired using different pathways (NHEJ and HR) during which editing happens. See these posts for more details:  &nbsp;&nbsp;&nbsp;&nbsp; How is the type of genetic manipulation determined in CRISPR-Cas9?  &nbsp;&nbsp;&nbsp;&nbsp; Mutations/deletions with CRISPR  &nbsp;&nbsp;&nbsp;&nbsp; How does NHEJ cause indels?
It is also possible to edit DNA in a way that does not involve cutting of the DNA. These techniques use similar principle as in the case of RNA editing described in the paper that you linked (see the abstract and introduction of this paper). However, if you cut the RNA, then it will be completely degraded instead of being repaired.