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.

In spirit they are different things. But the usual convention is to think of vector in the plane or in three-dimensional space as starting at the origin. In that case, a vector is identified precisely by its ending point, giving you an identification between points and vectors. 
One way to see that they are different things (even if identified in many circumstances), is that you can add vectors, while the sum of points makes no sense. Same with the dot and cross products. 
