You can remote play PS4 games on your PS Vita even without a wireless router. SCE President of Worldwide Studios Shuhei Yoshida tweeted to a fan that the system can directly connect on top of the usual networking solution.
@joshread Once you had registered them, yes. And even if the router is down, too.— Shuhei Yoshida (@yosp) October 31, 2013
The Wii U Gamepad uses a similar function for its off-screen play. But the Gamepad relies solely on its direct connection with the Wii U system, whereas the PS4/Vita combo can connect directly or over a network--meaning its range will likely be better, though going through a router would entail some extra latency.
Sony recommends a wired internet connection for PS4 users who desire the best remote play experience possible. Remote playing PS4 games on a PS Vita outside of its home network "may or may not work."