Our definition of separation is:
If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and we cannot have that $\{x_n\} \in A$, $x_n \to x$, with $x \in B$ (and also vice versa).
And the professor says that separation is not the same as if there exists $r>0$ such that $d(a,b) \geq r$, $\forall a \in A$, $\forall b \in B$. What is an example of a non-separated set (a set that fails to be separated by our definition), but meets this condition?
Your professor probably means that the other condition is sufficient, but not necessary, for a space to be separated.
If that condition is satisfied, then the space is separated by the sets $$\bigcup_{a\in A}B_{r/2}(a) $$ and $$\bigcup_{b\in B}B_{r/2}(b)$$
However, removing any single line from the plane leaves a separated space that does not satisfy this condition, since there are points from each of the two remaining half planes that are arbitrarily close to one another.
In other words, (condition) $\Rightarrow$ (separated), but (separated) $\nRightarrow$ (condition).