Admittedly, that is a complicated question, going back to a time before either of those amalgamated tribes was formed, but it seems that the biggest difference came to me in a flash while I was at the Calumet County courthouse yesterday.
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Stockbridge, Wisconsin and Brothertown, Wisconsin are Calumet County villages that were once populated by the "civilized" New York Indian tribes bearing those names. Today the Stockbridges (also known as "Mohicans" or the Stockbridge-Munsee Band of Mohican Indians), have a federally recognized tribal government and reservation in Shawano County, Wisconsin. But the Brothertowns are just doing their best to stay organized. They aren't recognized by the federal government, but, thanks to their efforts, they have a relationship with Calumet County (see Darren Kroenke's comment below).
The Calumet County Courthouse holds two record books. One for the Brothertowns from 1839 and one for the Stockbridge Mohicans from 1843. Each record book does the same thing for its respective Native nation: it allots their reservation, that is, each record book documents the dividing up of what had been a reservation into individually-owned plots of land. With these record books, each tribe was giving up their tribal government and becoming citizens of the United States.
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The two record books, of course, establish that both tribes went through the process of allotment at roughly the same time.
So what was different?
The whole time the allotment process was happening, the Stockbridges had an "Indian Party," made up of some of their leading men: John W. Quinney, Austin E. Quinney, and John Metoxen. Although the Stockbridge record book of 1843 makes no mention of any objections to what the elected commissioners (members of the Stockbridge "Citizen Party") were doing, John W. Quinney had contacts in Washington D.C. and that is where he took his fight - and won. In 1846 new legislation was enacted that re-created the Stockbridge nation and reservation.
There would be more challenges ahead for the Stockbridge Mohicans, but they dodged an awfully big bullet in the 1840's.
In my opinion, Congress' Act of 1846 - and the Stockbridge Indian Party's ability to make it happen - is, more than anything else, the big difference between the Stockbridge Indians and the Brothertown Indians.