What is a flow through the cut in the Ford-Fulkerson algorithm?

Question

In page 12 of the slide, it states flow across a cut $(S, T)$ is $f(S, T) = \sum_{u\in S} \sum_{v\in T} f(u,v) - \sum_{u\in S} \sum_{v\in T} f(v,u)$.

I think the first part $\sum_{u\in S} \sum_{v\in T} f(u,v)$ is the amount of flow entering the network at the source node. The second part $\sum_{u\in S} \sum_{v\in T} f(v,u)$ is the amount of flow leaving the network at the sink node. Shouldn't these two be the same? Why the flow across a cut is the difference of these two?