I checked 20-some-odd sites which attributed this to Twain, and I find it suspicious that none of them gave an actual source.
It appears virtually certain that another Weather quote frequently attributed to Twain, "Everybody talks about the weather but nobody does anything about it", in fact was originally said by his friend and sometime collaborator Charles Dudley Warner.
Twain did, however, speak at great length about the variability of New England weather, in a speech delivered at the New England Society's Seventy-First Annual Dinner, New York City, Dec. 22, 1876.. He begins
I reverently believe that the Maker who made us all makes everything in New England but the weather. I don't know who makes that, but I think it must be raw apprentices in the weather-clerk's factory who experiment and learn how, in New England, for board and clothes, and then are promoted to make weather for countries that require a good article, and will take their custom elsewhere if they don't get it.
There is a sumptuous variety about the New England weather that compels the stranger's admiration -- and regret. The weather is always doing something there; always attending strictly to business; always getting up new designs and trying them on the people to see how they will go. But it gets through more business in spring than in any other season.
In the spring I have counted one hundred and thirty-six different kinds of weather inside of four-and-twenty hours. It was I that made the fame and fortune of that man that had that marvelous collection of weather on exhibition at the Centennial, that so astounded the foreigners. He was going to travel all over the world and get specimens from all the climes. I said, "Don't you do it; you come to New England on a favorable spring day." I told him what we could do in the way of style, variety, and quantity. Well, he came and he made his collection in four days. As to variety, why, he confessed that he got hundreds of kinds of weather that he had never heard of before. And as to quantity -- well, after he had picked out and discarded all that was blemished in any way, he not only had weather enough, but weather to spare; weather to hire out; weather to sell; to deposit; weather to invest; weather to give to the poor.
That's plenty of context if you ever verify the attribution.