I've done the JMT in June, July, August, September and plan to add a few more months to that list.
Of the above months, my favorite for weather, bugs, water availability is August, but very crowded, very dusty, lots of smoke from fires possible, no snow, no flowers, scenery getting bland compared to July. Also, days are shorter, which for a high mile hiker makes a difference over July.
July probably the best all around, especially in high snow years. Fewer people than in August, long days great flowers and scenery, much more water (sometimes almost too much), and great alpine feel with still some snow around. Skeeters are on and off - some areas they are just insane, then in the next valley you won't have any. Last year, Bear Creek area was the only place we had real bug problems, no bugs north of there, almost none south. Weather can be unsettled, although even in a strong monsoon year, we usually only hit one or two rainy (or snowy...) days.
September - my least favorite so far. Short days, dull, brown scenery, getting very dry so you have to haul more water. No more flowers, tons of people on the trail who want to "avoid the August crowds" - I did a northbound hike in Sept and saw on average 100 people a day. Compare to about 20 southbound in any July (except Mt. Whitney trail where we saw 220 on one day). Weather stable, though.
June - in a dry year, you can do the trail in mid June, and you won't see many people at all. The more snow and deep crossings you can handle, the earlier you can start and the more solitude you'll find (except for those pesky PCT northbound hikers who come through rather early as well), but the more hard core the hike gets, especially on Mather and Glen, where snow can get you into sketchy situations if you don't know the exact trail routing and climb into the wrong slope. Almost too early for bugs for most elevations, but thick near the lower valleys. Weather much more stable than in July, but when it turns, it can still snow sometimes.
