They are two very different types of colleges.
Vanderbilt is a large private research university, with 7,000 undergraduates and 3,000 graduate students (PhD and masters). Faculty split their time between teaching and research, and split the teaching between graduate students and undergraduate students. There are more faculty, and a more diverse set of topics in the field
It’s in the South, and it’s in a large city, Nashville, Tennessee. Nashville has around 700,000 people, and there are around 1.9 million people in the urban area. The weather is generally hot and humid in the summer, warms and humid in the Spring and Fall, and cool and dry, sometime cold and dry, in the winter.
William is a small Liberal Arts College, with 2,000 undergraduates and some 50 masters students. It is focussed almost entirely on undergraduate education, and faculty teach undergraduates and their primary job description, followed by research. With fewer students, and therefore fewer faculty, with fewer topics.
It’s in New England (the Northeast), and it’s in Williamston, a small town of 7,500 people, next to a South Adams, a slightly larger town of 13,000 people. The closest city is Albany, an hour’s drive away. The weather is warm in the summer, early fall and late spring, cool to cold early spring and late fall, and very cold and snowy in the winter.
Summary:
Vanderbilt: large, many students, more faculty, but faculty have less time for undergraduates. Hot and humid to cool weather, largish city, in the South.
Williams: small, few students, faculty have more time for undergraduates, but there are fewer faculty (so fewer topics). Ward to very cold weather small town in rural area, in the Northeast.
I also recommend that you read up a bit on the differences between private American Research Universities and American Liberal Arts Colleges. They have few counterparts in other countries.