Catholic/protestant divide?
Edwards represented CD7 and relied on the Cajun vote. Johnston was from Shreveport so he cleaned up in the north. Johnston also ran on a good government, "throw-out-the bums" platform; thats why you see him doing well in Baton Rouge and (narrowly) winning the New Orleans area. My guess is that Johnston would have won New Orleans more decisively, but Edwards voted for civil rights legislation in Congress, which helped him with blacks.
This was one of the reasons Edwards created the jungle primary. This was when LA used the traditional system. Edwards got into this runoff after getting 23% in a 14-way primary (this map is actually the runoff, not the primary, my mistake). Then had to run in this tough race plus pivot to the general election against Treen, a (relatively) good Republican recruit. Too many hard elections for one contest!
But funny that if Johnston won this race in 1971 it could have ended Edwards' political career...