Generally speaking the earlier in the season the better the weather and the longer the day length. Also, the top rams haven't been picked over. Later in the season you may have the country to yourself and snow may drive a few rams out of their hidden spots. I've been on 2 dall hunts in Alaska and always went opening. Winning a Tok tag is like winning lotto. It won't matter when you go you ought to have an incredible experience!
When I hunted mine in the Yukon, I went in late Sept. because I wanted a 'haired up' ram whose neck would look like a linebacker! We had snow, but it didn't really hinder us. Took a 14 yr. old warrior.
I've hunted sheep for 13 years in Alaska. The opening week or so is almost always much wetter (snow and rain) than say the last week of August and into September. I prefer late August into September because of the drier weather, cooler temps, better hair length on the cape, and fewer sheep hunters in the field. With that said, there's no doubt that some of the legal rams have been harvested by the time I get out which is a real downer in heavily hunted areas. I've never been holed up in my tent for more than 1 day whereas early season hunters sometimes get rained/fogged out for their entire hunt. Clearly there are pros/cons with when you go.
My fault in regard to later weather being less favorable. I guess it depends upon where you are in Alaska and the particular year. I hunted mtn goats 2 years near Seward. The later into August/Sept the worse the weather was in that particular area. In fact, a lot of guys with boats put their boats away by Oct because it can get mighty dangerous! If you want to know the average weather info for the particular area you are interested in hunting go to the usclimatedata website. Here are the averages for Fairbanks: exacthttp://www.usclimatedata.com/climate/fairbanks/alaska/united-states/usak0083