How does private benefit exceed social benefit?
The net social welfare can be written as $W=B-C$, where $W$ is total welfare, $B$ is the benefit, and $C$ is the cost.

Suppose we are interested in thinking about a negative externality that implies a social cost of $E$. Then we would have $$W=B-(C+E)=B-C-E.$$

Imagine if, instead of viewing this as a social cost, we interpret the externality as reducing the social benefit. The expression would then be $$W=(B-E)-C=B-C-E.$$

So the two ways of thinking about the world are completely equivalent and it is merely a naming convention whether we call the externality a cost or a reduction in benefit. I guess your confusion has come about because you heard people on different occasions using the different conventions, without being clear about which convention they were using.

The only thing that matters is that we don't double-count the externality by including it as both a cost and a reduction in benefit.