In the realm of problem$-$solving agents, what distinguishes a 'depth$-$first search' from a 'breadth$-$first search' in handling large state spaces?

$$ Depth$-$first search uses a heuristic to guide its exploration

Depth$-$first search tends to use less memory but can get trapped in loops

Breadth$-$first search provides a more detailed problem description

Breadth$-$first search operates without a transition model