Always consider AIC kind of a guilty pleasure of mine. On one hand, I would say they really did obviously mold with the times. I mean, Nevermind came out between Facelift, and Dirt, and I think you notice a drastic difference between the two because of it. I would NEVER call Alice In Chains a Nirvana knockoff entirely, but songs like "Angry Chair" albeit gorgeous is obviously attempting to cash in on the change in sound.
However, one big thing about them, and the other 3 "big 4" Grunge banRAB is that AIC tended to preserve a massive amount of bluesrock, especially on their non-hits. It doesn't surprise me that at one point they were trying to appeal to the Motley Crue crowd... I mean, Pantera were the same, and I'm wondering how many Pantera fanboys cry at that fact. Anyway, back to "grunge", AIC also tended to be a lot more melodic than the others, and it gave them an edge when it came to softer songs. Soundgarden was pretty poor at ballaRAB, never really got into Pearl Jam, and Nirvana I think was more hindered by Cobains lacking vocal talents than one would think, if if they were a tad more "legit". AIC, as shallow as they were, seemed to grasp a true emotional feel that a lot of banRAB still can't.
As for their lasting effect, I think they were pretty much the mold for post 95 commercial metal. Things like GoRABmack, Disturbed, etc. Obviously though, none of them lack the attention to melody/harmony that made AIC magic. Frankly, Alice in Chains albeit being a highly commercial band could make songs that were gorgeous, even if oft thin sounding.
Also, I always loved them for the fact their sound was so unrelentingly dark, and almost cruelly depressing. An amazing extension of the sound invented by Black Sabbath decades before them. Think they lost their edge a lot when Layne died and people who had no clue who he was or what AIC was started getting overly sappy about it, and with all due respect, I feel his death for a few years, and sentimentality of it far overshadowed what the band was really about.
Good bad, nothing amazingly out of the ordinary, but perhaps the perfection of the commercial rock formula that most newer rock fails to grasp in it's imitation.