How well do you shoot other rifles? Bases on some of your questions you sound newer to the sport, no offense. But before I would look at something being wrong with the gun or ammo (a Mini is chamber spec'd for 5.56 but shoots .223 fine, and have a 1:9 twist which is ideal for 55gr ball.) I would assume it was the shooter, as is often the case. Mini's aren't known for good triggers, or sight, or stellar accuracy anyways. Couple that with an inexperienced shooter, I would presumably expect the same results.
I am not trying to offend you in anyway at all, but this is often the case we run into when people bring guns back for repair due to poor grouping. Even more so with handguns.
The short answer to you question for 5.56 vs .223 is NO... the long answer... yes there is a difference, however it would not affect your rifle because it is chambered properly. Guns chamber spec'd for 5.56 have a longer lead in them, basically for full auto and suppressed operation. 5.56 spec ammo runs at a little bit higher of a pressure, the only difference you would really notice is the way the brass ejects. I have never seen any huge accuracy difference good or bad switching between two similar types. Now when you start throwing match grade ammo vs the cheap bulk pack stuff is different.
The major concern with the different chamber spec'd are to NOT use 5.56 ammo in a gun that is chambered specifically to .223 Rem. 5.56 chambered guns is FINE like your Mini, to run .223 ammo. It is recommended not to run 5.56 ammo through a .223 spec'd rifle, however it has been VERY HARD to find any reports of failures due to it.
The American Eagle 55gr is more accurate than people give it credit for. I through a 12x Scope on my BCM 14.5'' mid upper to test some loads one day, and at 100 yards I could consistently group under an inch with the 55gr AE bulk pack. This does not mean your Ruger likes that ammo ofcourse.