Are you asking about a JMT northbound start, or just hiking around to do Whitney from the west?
June 17 there will be more water, but this year not a problem due to it's being such a dry winter. Maybe a few more mosquitos on the JMT.
July 16 is later, closer to fire season, more thunderstorms than in June. Cottonwood pass is less elevation to climb in the first/second day, compared to Cottonwood Lakes.