Instead of Reddit being a division on the Org. chart of C.N., Reddit Inc. shares are held by C.N. This means that Reddit can take on equity investors to finance operations and operate externally of any organizational influence Conde Nast had (which may not have been much anyways).
Reddit, Inc., will be owned by Advance Publications, which also owns Condé Nast, so it will be a corporate sibling of CN and more independent of it. Advance Publications owns other divisions in addition to CN, including a newspaper division that presumably does not is not influenced very much by the magazine folks: