This comes down to the difference between pathways and networks.

Pathways:

• schematic of biological knowledge
• indicates many different types of interactions
• tend to be human curated, hence tend to cover only a fraction of the genome
• e.g., KEGG, Reactome

Networks:

• a set of bimolecular interactions represented as a graph (directed or undirected)
• can capture well-studied and not so well-studied biology
• much higher coverage than pathways, but supported by less data
• e.g., BioGRID, IntAct

Pathway analysis uses more complex models to try to utilize the additional mechanistic and regulatory information that pathways contain. Network analysis usually means overlaying a set of genes on top of a pre-existing network and asking whether they cluster in some region.

I would add this slightly nuanced definition:

Pathway analysis is analyzing a list of genes against an a priori set of pathways with known interactions for various systems, and identifying which pathway is most impacted.

Network analysis is analyzing how a set of genes are connected given a known set of direct interactions, and identifying which genes cluster in a network.