It sounds like you're looking at formulating a company-wide policy with respect to use of cookies in web application development.
As such, for a company-wide policy, be careful to consider not only your typical type webapp where server produces HTML+JavaScript, but also any potential web APIs that company web applications may be publishing. Such web APIs may be for AJAX purposes, but also may be for consumption by other type clients, for example B2B type data feeds, that may rely on some form of persistence on the consumer end. For example a "browser" like Twilio only understands TwiML, as opposed to HTML+JS, and local storage is not applicable there. And if webapp that interfaces with Twilio relies on persistent storage, local storage is not an option (whereas cookies are).
This is not to say that such applications, if such exist or will exist in your organization, cannot be (re-)designed to avoid need for client-side persistence. This is to say that local storage may not necessarily always be available in all contexts to provide alternative to cookies.
Otherwise, user2428118's answer nicely contrasts the two technologies.