Both are equally valid approaches to hearing aid verification, and historically are associated with the philosophies of prescribing gain according to the well-known NAL (REIG) and DSL (REAR) prescription methods for hearing aid gain.
One of the most commonly cited advantages of the REIG is that it is a relative measure; the difference between REUR and REAR. This means that precision probe-tube placement is somewhat less critical than an absolute measure (REAR), so long as the probe-tube does not move between measurement of the two steps. On the other hand, such an advantage is at the same time a hindrance; if, as part of the REIG process, one needs to measure the REUR and REAR, why not skip the REUR step and concentrate on the REAR? Using REAR would also imply fitting and verifying the hearing aid using the SPLogram (rather than showing hearing aid gain). This arguably makes visualisation of the patients dynamic range easier. This point is important both for clinicians (who must not only match the hearing aid to target, but also try to ensure audibility across the frequency range for different input levels, while bearing in mind the uncomfortable loudness level) and patients (who are often involved in the hearing aid fitting and counselling process via speech mapping and other visual tools that based on the SPLogram).
There is much literature covering this topic and it is something of a long-standing debate amongst experts in fitting hearing aids. For a brief review please see Jorgensen (2016).
Jorgensen, L.E. (2016) Verification and validation of hearing aids: Opportunity not an obstacle. Journal of Otology, 11, pages 57-62