What is the difference between a point and a vector? I understand that a vector has direction and magnitude whereas a point doesn't.
However, in the course notes that I am using, it is stated that a point is the same as a vector.
Also, can you do cross product and dot product using two points instead of two vectors? I don't think so, but my roommate insists yes, and I'm kind of confused now.

There is a difference of definition in most sciences, but what I suspect you're asking about is a rather nice one-to-one correspondence between points in real space (say perhaps $\mathbb{R}^n$) and vectors between $(0, 0, 0)$ and those points in the space of $n$-dimensional vectors.
So, for every point $(a, b, c)$ in $\mathbb{R}^3$, there's a vector $(a, b, c)$ in the space of all 3-dimensional vectors.
