The password correct, sometimes it works sometimes it doesn't

Hi,

I own an iPhone 6 Plus, and I've been having problems with my accounts (App Store and iCloud) and it's driving me crazy.

When I download a new app from the App Store, it works fine, the password works fine.

When I want to update apps from the App Store, just nothing happens, nothing.

It's from the same Apple account, same e-mail, same password, just the apps don't update.

I tried resetting my password (and I did many times), still the same problem.

When I go to the iCloud (on the iPhone), when I sign in, they ask me for the password, and I put it, and I get a V saying it's right.

When I go to sign out, they ask me again for the iCloud password, and I put the same password, and it doesn't work, it says the password is wrong.

I went to the iCloud website, logged in with the same account that I have on the phone, and it works.

I tried to use the "log out from all devices" from the website, and it logs out from everywhere aside from the iPhone iCloud.

I'm losing it, I can't do anything, I keep getting the notification from the iCloud that I need to sign in, and the password doesn't work.

Does somebody know what is happening? I'm going nuts! It's only a 2 months old iPhone 6 plus!

Thank you