A semantic network is mainly a way to represent knowledge as a graph. An ontology goes further: it gives that graph a formal specification of what kinds of things and relations are allowed and what they mean.

For example, a semantic network might contain:

Socrates —IS-A→ Human
Human —IS-A→ Mortal

That already represents useful meaning through nodes and links.

An ontology would typically define things such as:

Human is a class.

Person is a broader class.

Human is a subclass of Person.

parentOf is a relation between two Persons.

hasMother can have at most one value.

parentOf is the inverse of childOf.

So the ontology describes the conceptual system itself, not merely individual facts.