The tourism operators aim to raise their rates for the tourist season on June 1, but because the traffic always isn't quite there yet, they usually end up fighting over the smaller numbers of tourists and keeping rates low until about June 10.

I did a search for a weekend rental car June 6-8 (picking up AM and returning PM) and June 13-15 (same thing), since that's the part of the tourism industry with which I am most familiar.

Cheapest car for the first weekend: $192 all-in
Cheapest car for the second weekend: $320 all-in

I would assume hotels have a similar spread (my wireless router's being fussy and dropping packets, or else I'd do a random check now). Factor the price of the things up here on top of the airfare--the later weekend might break the bank.

As for the disadvantages of going the first weekend:
-The weather will likely be warmer the second weekend, although there is always the chance we'll have 75 degrees the first weekend and it will rain and be 60 the second weekend--but in general, the later in June you go, the better and warmer the weather
-The earlier in the season you go, the less time the roads have had to dry out--important if you're going to Denali, as the park shuttles may not operate the full length of the park road until it's dry

You'd think a week might not make much difference, but there's a reason the prices are higher the second weekend and there are twenty times as many tourists in Alaska the second weekend. Spring and summer hits fast and hard here--a week and a half ago it was 25° and snowed 22 inches, and then this weekend, it was 55°. I don't think you'll have particularly cold or bad weather the first weekend, but the chances of it being warm and sunny and the park road (which is dirt) to be completely dried out and everything open for business is much higher the second weekend.

Then again, some people like the shoulder season for its thinner crowds--you might want to take that into consideration...