The best way to Remote Play at the moment isn't on the Vita, it's on the PS4 Remote play app available on Android and PC/Mac. Why? Because of the poor 2.4Ghz legacy Wifi standards, even on the new PSVita Slim. I've tested this recently with Persona 5, playing it on the same spot using both the PS Vita Slim and the PS4 Remote Play app on Windows 10 laptop. On the laptop running on 5GHz WiFi, smooth sailing. On the Vita Slim with its 2.4Ghz WiFi, it'll eventually stutter and loses connection. With the proliferation of wifi hotspots and Bluetooth wireless devices that uses the same 2.4Ghz band, your Vita Slim chokes when it tries to compete with other devices that transmit and uses the same 2.4Ghz band. My case applies to using PS Vita outside the house on mobile broadband connection. I used the E5786 MIFi fed by DiGi Infinite 150 data plan. Don't even hope of running Remote Play using free hotspots like the ones at Starbucks, they're crappy in upload stream quota, your Vita needs 5Mbps up/downstream connection speed to do Remote Play adequately.
I tested this while being stationary, like chilling out at coffee houses and eateries, and I've also tested this while traveling inside the LRT on the Kelana Jaya line. The LRT experiment was the most extreme case and naturally it stutters and cuts off underground, but on above ground areas with ample coverage, it works but is always in danger of getting disconnected. But because I'm playing Persona 5 streamed via Remote Play to the Vita, and P5 isn't a twitch heavy, fast paced game, it isn't a big deal if Remote Play loses connection, since the game will wait for your command once you reconnects and if you can't, Rest Mode will take care of the rest.
On your home network, you don't even need a WiFi network to Remote Play, you just need your connect to your PS4 directly via Direct WiFi (PS4 creates it own wireless network and your Vita logs on into it) , but of course, if you want expanded network coverage, a dedicated router helps. Home streaming is easy, real Remote Play streamed remotely outside of your home network coverage is the one tough to tackle.
I've spent about 50 hours of my 310 hours playthroughs of Persona 5, playing on Remote Play between my Vita and my PS4 Pro. The most important thing people always forget when they wanna Remote Play is that, don't do a double wireless connection: connect your PS4 to your router via wired connection, that way, you minimize the packet loss due to wireless transmitting to the absolute minimum. Ultimately, if you're Remote playing outside of your home network, your mobile broadband (or outside wifi/network coverage of your choice) will determine if you're having a good time on Remote Play. I have a mixed experience, sometimes good sometimes horrible.
Thanks for sharing the experience. Glad to know that it's possible to enjoy PS4 on the go via Remote Play.