For 50+ years, the LL age cutoff was July 31st -- I think that is the date Branson Baseball is remembering. That date was chosen at the very beginning of LL because the school district in Williamsport, where LL was founded, used that as its grade cutoff. The idea was to have kids play with the kids in their grade.
In 2006 LL changed its date to April 30th, based on a request from USA Baseball. Press release at the time stated:
WILLIAMSPORT, Pa. (May 9, 2005) – The Little League International Board of Directors, governing body for the world’s largest organized youth sports program, has accepted a recommendation from USA Baseball to change the league age determination date for its players starting in the 2006 season.
The league age determination date is the age a player has attained as of a specific date, for the purpose of placing the player in a particular division. For more than 55 years in Little League, that date was July 31. (see, for example: www.southamptonlittleleague.com/news.ihtml.html)
I have heard that USA Baseball was really pushing for a Dec. 31st date, to align with international baseball, and that LL agreed to move to that date in two steps. The second step was later announced, and planned for 2018. Of course, the unintended consequences of moving to April 30th was that 13 year olds started dominating the tiny LL fields. Much easier to change the date back to late summer than to reconfigure thousands of fields. IMO, late summer (whether July 31st or Aug. 31st) has always made the most sense.
We have LL in my town, but Pony in a neighboring town. As for Pony vs. LL, I don't think it makes a bit of difference what baseball a 7, 8, or 9 year old plays. By 11 or 12 the kids who are most interested in baseball are playing some amount of travel -- whether or not their town offers Pony or Little League.
(Edit: I didn't see Midlo Dad's post -- he covered the same stuff.)