Fundamentally, Gracie Jiu Jitsu and BJJ (Brazilian Jiu Jitsu) are the same martial art. Some members of the Gracie family such as the Gracie Academy like to emphasise more self-defence elements, particularly in stand-up whilst many modern exponents focused on BJJ competitions such as Gracie Barra have taken the delivery system of Gracie Jiu Jitsu (ground fighting) and added other arts like Judo or wrestling for BJJ competitions (see Guerrilla Jiu Jitsu by the Camarillo Brothers) whilst others have incorporated it into MMA with modifications for no-gi like 10th Plant whilst others such as SBG use BJJ in all 4 elements; Gi, No Gi, Self Defence, and MMA. In practice, the most important factors for someone to consider when looking to start Gracie Jiu Jitsu or BJJ is not the name itself, but that the club concerned is led by coaches with a reputable standard in the art. The lineage of the instructor should be clear, transparent and lead directly to Helio Gracie, Carlos Gracie, Luis Franca, or Oswaldo Fadda. This may seem pedantic but a weekly occurrence on Facebook BJJ groups is the outing of unqualified traditional jujitsu instructors making false claims of holding BJJ Black Belts that they simply don’t have. Their students are being misled, wasting years of training and money only to discover that what they have learned in valueless. They often discover this when they happen to train at a real BJJ club and they are repeatedly submitted by white belts despite holding supposedly high grades under their fraudulent BJJ instructor. Whilst some move on and start again, many leave having never really started.
In terms of self-defence, training at an authentic BJJ club will give practitioners better self-defence on the ground than any other martial art offers, even if the club is entirely focused on BJJ competitions. The notion that someone like Rafa Mendes, a world champion in BJJ, or any of his students would be unable to defend themselves effectively in a street confrontation is utter nonsense as well as misleading.