You do recall all the interviews and commentaries where Matt Groening said he specifically wanted to make sure The Simpsons was NOT like The Flintstones, right?
The biggest influence The Flintstones had on The Simpsons was merely that they were both cartoons in prime time. And in meetings they sold it to FOX as "The Flintstones for the 90s" even though they had no real intention of making it like that. But Groening has talked quite a bit about how excited he was for a cartoon to be airing in prime time as a kid, and then being disappointed with The Flinstones' lameness. The entire goal was to make a show much better than the garbage Hanna Barbera had spewed out. (Similarly, with Futurama, part of the goal was not to make it "bland and boring like The Jetsons," which FOX executives objected to because they apparently loved The Jetsons.)
Calling Homer's best friend "Barney" as a gag aside, most of the similarities between The Simpsons and The Flintstones are merely superficial ones that came about only because both shows were mining from some of the same larger tropes. Such as from The Honeymooners, for instance. The difference is, The Simpsons was creative and funny and really did something different with those tropes, whereas The Flintstones - much like Family Guy - couldn't really be bothered to.