If you think they are not, please give an example for a string that can be called "URL" and not "path" and one that can be called "path" and not "URL".
URL: http://stackoverflow.com/questions/3491674/are-the-terms-url-and-path-interchangeable
Path: /questions/3491674/are-the-terms-url-and-path-interchangeable
Schema: http
Host: stackoverflow.com
They're all different parts of the URL, they're not interchangeable.
URL is mostly a "web specific term", but path can be used in a lot of different contexts, filesystem path, XML [x]path, etc.
Getting if from the source: RFC 1738
An HTTP URL takes the form:
http://<host>:<port>/<path>?<searchpart>
A URI (including URLs though the distinction is irrelevant in most cases, and URLs are always URIs so mostly we just talk about URIs these days) is a globally-unique identifier of a resource.
E.g. http://example.net/foo/bar?a=b
A path is part of the URI, in the above example the path is /foo/bar
A URI reference is a means to textually communicate a URI. It can be absolute, in which case it is the same as the URI, and hence it will resolve to the same URI in any context. It can be an absolute-path, like /foo/bar?a=b in which case it will resolve to the same URI in any context sharing the same scheme, host and port. It can be relative such as ../bar?a=b and it can be scheme-relative such as ://example.net/foo/bar?a=b which resolves to the example URI above from any HTTP base URI, but not if the base URI is HTTPS or another scheme.