About the only thing I really want to have is an app that does internet radio streaming, i.e., TuneIN. Or, given that I only listen to trance music, a Digitally Imported app. I've seen a few people via twitter and forums request these apps from the respective companies but so far, there's been no response yet.
For the browser, I personally think that clicking on a video should open some sort of video pop-up within the browser that would perform the playback.
The PS store could use some work. I like to see video previews* and a wishlist in there. The 3DS eShop already has both.
*We shouldn't have to download previews just to watch them.
I agree with the PS Store needing some work. The huge PS Store overhaul on the PS3 would not only look great but work even better on the Vita because of the touchscreen. I don't think much would need to be trimmed to make it work on the Vita. Besides, having the same PS Store on both the PS3 and PS Vita provides for a consistent experience across both consoles.
Another thing I'd like to see on the Vita would be increased cross conctoller functionality with the PS3's system software and apps. For example, I'd like to control the PS3's XMB from my Vita and use my Vita as a keyboard things like the PS Store, Netflix & Youtube, to name a few use cases. The Vita could connect to the PS3 via Bluetooth just like the Dualshock 3 controller to avoid any latency issues of a LAN connection.
One time I was on skype on the vita and and error sign popped up I closed it and it sent me back to the vita home screen but my friend was still talking!! I could go on anything as if i wasn't even on skype and still talk to my friend...i know there is already Party chat but this would be awesome...Also ever since that latest update the group messaging is messed up everytime I try to send something its sent straight to me on my ps3....idk why..