Pretty much as above. I have download both the free apps Facebook and flashlight to get me started on my new phone.
I first down loaded onto the comp however it says my computer is not authorised to sync with these? Which is weird cos it is my phone and comp. So i then download from my phone, where they both worked initially but now don't work again since I hooked up with the comp where again it tries to sync and put on the computer but again with no luck. Now the apps on the phone wont even open. Anyone have any ideas?

Double click on one of the Applications in iTunes, this will prompt you to authorize your computer. That should fix it, next time you sync your iPhone with iTunes the Apps will go across and work.

Hrmm I have had a similar issue with my iPhone Classic in that I try to download apps from the Wi-Fi App Store and it says that it has failed. But then next time I sync to iTunes it downloads from iTunes Store.

Do the apps show up in your application tab in iTunes?

I'm having an issue where I will download apps on the iPhone but they will not sync with iTunes and show up under 'Applications' although the apps I download via iTunes are there.
Anyone else having this problem?

Yes. And when I try to sync them across I get an error (-4 from memory).

Ok they are now syncing with the phone as I did the double tap which asked me to authorise so its syncing with the phone. The faceboof application however is still not opening on the phone? Is there a way to remove the app on the phone so i can try again?