I understand why this is a tough choice. Both are great schools.
One warning: I have a degree from Stanford in a math-heavy subject, which might make me a bit biased. However, even with this I would be leaning towards Williams College. Here is how I see it.
I did like the quarter system at Stanford. Classes go fast. The end of the quarter comes up quickly. Then you get a new set of classes next quarter. I am assuming by “non-traditional” you mean a little bit more mature, which to me fits well with the quarter system and the associated need to keep ahead in your work.
The math courses at Stanford are not easy in my experience.
In terms of the name recognition, I agree that if you walk into a bar somewhere in the middle of America Stanford has a much stronger name recognition. However, you are in a field where graduate school looks very likely. For applying to graduate schools, I think that both are excellent and both are very highly regarded. Graduate school admissions, including at Stanford, will know how strong Williams is.
I have lived in California, I have lived in Massachusetts, I have lived in Montreal. I think that you will get used to the cold of western Massachusetts. While I do see the advantage of Stanford weather, I think that this really doesn’t matter one way or the other.
“Will have to work during the school year” seems unfortunate to me. Either school would require quite a strong focus on academics. To me “I will end up paying close to nothing to attend which allows me to take the time/energy I need to focus on my studies” looks big. How many hours you would need to work to attend Stanford may matter, but given the cost of education it is hard to see how you could earn enough money to matter without working more hours than I would have been comfortable working while still being a full time student at Stanford.
“Allowing me to study abroad” to me looks good also.
“Has been my dream school” and cost IMHO seem like the biggest issues when deciding between two somewhat different but both exceptional options.
But I do not think that you can go wrong either way, as long as you would not end up with debt to attend Stanford.