PSP is still selling well because of some key factors. PSP is cheap and so are it's memory cards, it can be hacked, play homebrew games, emulators, and has a big library of quality titles since it's an old console. Vita doesn't have any of that for now. I'm not saying it'll never sell well, but comparing PSP's situation with Vita's is not the right way to make a point.

Yeah I am not a a handled player but if this Vita remote play is true. Then I am buying the Vita at some point in tims

Remote play has been around for years on the PSP and PS3. You could turn on your and access certain games, music, videos from your PSP from any WiFi connection anywhere. It's an awesome feature, the only problem is it is unreliable. Sometimes it doesn't work at all and when it does its spotty at best. Games would have lag and the PS3 is tied up while remote play is in progress. Having the wrong type of router could even cause issues with it. Even when directly connected to the PS3 through its WiFi adapter it wasn't perfect. Remote play was one of the biggest reasons I got my PSP but it just never panned out.

So far everything I have read hints to remote play being limited to your home WiFi for Vita PS4 or at least that's how they are pushing it since that is were you'll likely get the best performance. But if with GaiKai they do a better job with the game streaming and continue with cross-buy/play/save, and they improve the remote play functions for this gen I might actually try again with a Vita.

It won't save the Vita per say, but it is definitely a nice bonus feature, one that I will try to extensively use.

Remote play via internet? Well, this requires your ps4 to be always online, always connected. On top of that you need a decent internet-access. Given most consumer-internet is asymmetric you better look for a good access with high upload. On top of that you better have a low latency at your access. On top of that nobody should use your internet-connection while ps4 is streaming. On top of that ps4 will have to leave low-power-mode and will consume a lot of power. On top of that if you play via 3g you won't have a good experience because of latency. On top of that it will eat up a lot of data from your limited data-plan. If you want to play via wifi at any remote location it will highly depend on how the internet-access is where you get your wifi from.
Yeah, sounds good :)

It works but has issues. And anything you do online depends on your connection. It's not restricted to the remote play feature. It does sound good but its whether it IS good in practice and the answer is it just depends. We'll have to wait and see what they have done to improve it AND what they are doing with Gaikai. Oh and the remote play feature on PSP/PS3 had options to lower the bitrates and latencies to better match your connection and streaming music and movies works pretty well.

As I said though it was spotty at best for games but with cross-play/buy/save that at least allows you to play games without the need to stream them when on the go. So it has promise just needs to be improved a lot.