There is still a difference. Once you submit, AMCAS verifies your application before sending it on. That can take a few weeks. If you submit on June 23rd, your application will not be ready to go out on the 24th. Instead it will go out in early/mid-July.
Now, how soon the adcoms get on those early apps is variable and unpredictable. The school could sit on the apps until August at which point there would have been no difference in early June versus early July submission.
But still, with rolling admissions you want to be early so why not submit as soon as the app is ready? If that's early June, awesome. If that's early July, great.