Handoff or tool, which one

Both put a second agent to work. The difference is who is left holding the conversation.

The same billing agent, wired both ways in one file, so the difference is the only thing left to look at.

by_handoff = Agent(name="Triage", instructions="Route it.",
                   model=PretendModel([call("transfer_to_billing")]),
                   handoffs=[billing])

by_tool = Agent(name="Triage", instructions="Ask billing, then reply.",
                model=PretendModel([call("ask_billing", input="refund"),
                                    "Billing say the refund has started."]),
                tools=[billing.as_tool(tool_name="ask_billing",
                                       tool_description="Ask the billing team.")])

for agent in (by_handoff, by_tool):
    result = await Runner.run(agent, "I was charged twice")
    print(f"{result.last_agent.name:<8} said: {result.final_output}")

Read the two lines carefully. In the first, Billing is the last agent and the words are Billing's. In the second, Triage is still the last agent and the words are Triage's, informed by what Billing said.

Choosing

A support desk is usually handoffs, because a refund question really should become Billing's problem. A writing or checking step is usually a tool, because you want the answer back.