June in the Rockies can dish out nearly anything, but it’s not all that different from the AT. At least I experienced plenty of volatile conditions on my AT thru–snow, violent thunderstorms, etc. I think the main difference from an AT thru is the higher elevation, so you’re more exposed, and you may have colder nights. Bail-out options are often more limited, too.
I didn’t carry any different clothing on my AT thru than I did on the PCT or CDT. But I hiked the AT last so I had a different learning curve than you. I imagine you’ll be OK with what you carried through the Whites. Conditions will likely be similar.
One difference I noticed between western and eastern hikers is that western hikers tend to wear rain gear when it rains. You’re more exposed to wind and temperature drops than in the summer storms back east.
In my two thru hikes starting in Glacier in June, I experienced a mix of heavy rain, blowing snow in the higher elevations, and some perfect days mixed in there, too. A lot like the southern AT in April.
Good luck and have fun.
Garlic