They start the same way. Both are a written brief with a job. A skill is the brief loaded into your own window, so the model you are talking to follows it, in your conversation. An agent is the brief handed to a separate window with its own model and tools, which does the job and sends back only the result. That separate window still inherits your CLAUDE.md, so it knows what you know at the start, without knowing what your conversation has been through. An agent can be given a skill to follow, so the two combine.
So the choice is simple. If the job needs to be in your conversation, use a skill. If the job would fill your window with stuff you don't want, or you want it sharp and unpolluted, give it to an agent.
Reveal on the spot: say whether you can invoke skills and start agents from here. Look at the conversation for something that was done in the main window and would have been sharper handed off. Name it.
Unlocks: knowing which of the two to reach for, without overthinking it.