Differential syntax for graphing

Can someone please clarify what the difference between "~" and "," are when separating x and y axis syntax? Does this simply have to do with whatever package I am loading+using at the time?

Also, in a similar manner what is the difference between the uses of "==" and "="?

Yes.

== is the equality operator to test whether two variables are equal. = is the assignment operator to associate some value with a name. It operates mostly the same as the <- operator. The <- operator is favored for assigning values to new names outside function definitions, and = is favored within.