The vulnerability is the exposed weakness. Vulnerabilities get fixes, and they exist without anybody knowing about them. Vulnerabilities get CVEs assigned to them.
The exploit is the means of attack. It's the specific actions or calls that let you take advantage of a vulnerability. It could be a worm, or botnet scripts, or specifically crafted data[0]. A proof of concept is not an exploit itself, but it demonstrates that the vulnerability can be exploited.
An example of a vulnerability might be a gate where the gap between the door and the jam are too wide. The exploit is a coat hanger used to lift the inside latch from outside the gate. That results in unprivileged access.
And zero-day specifically compares when the white hats (vendors, system owners) and the black hats learn about the existence of a vulnerability. If white hats learn that a vulnerability exists by being subject to an in-the-wild black hat exploit of it, then it's a true zero-day.