308 preserves the HTTP verb[1], and the form on that page uses POST. POST is not idempotent[2], which means using it more than once with the same parameters may not yield the same output. For example, POSTing this comment form twice would append to the resource twice; as opposed to GETing twice which just returns the resource unmodified both times.
Firefox correctly (under the old RFC for a 301 redirect[3]) asks for confirmation before automatically repeating a request that is not guaranteed to be safe to repeat. Some implementations will instead convert the request into a GET, which is why 308 was needed in the first place.
RFC 7238 with the 308 status code isn't part of HTTP/1.1 (even under the new revision, which is in RFCs 7230-7235), its an experimental extension that expressly notes that implementers must be aware that HTTP/1.1 clients not specifically written to that extension will fall back to the behavior for status code 300 when status code 308 is encountered, and that 308 should not be used where that behavior is not acceptable. (This is the standard mechanism for extensibility in response codes within the existing high-level groupings in HTTP/1.1.)
Section 6 of [RFC7231] requires recipients to treat unknown 3xx
status codes the same way as status code 300 Multiple Choices
([RFC7231], Section 6.4.1). Thus, servers will not be able to rely
on automatic redirection happening similar to status codes 301, 302,
or 307.
Therefore, initial use of status code 308 will be restricted to cases
where the server has sufficient confidence in the client's
understanding the new code or when a fallback to the semantics of
status code 300 is not problematic.
