Same symptoms - let's look at the "cause". I was digging in my iPhone settings and although I did everything as prompted in the iCloud setup here is what has actually happened under the hood.
First - What iCloud guided me to do.
•I had a @mac Apple ID I decided to use it for iCloud
•iCloud said I neeeded a new @me address to use mail and iMessaging and asked me to set one up
•I set up a new @me email address
Sencond - The Results
•I can receive email at my new @me address (expected)
•I can login to iCloud online with my old Apple ID (expected)
•I can not send outgoing mail - error on password (unexpected)
Third - Under the Hood (Settings that are a little odd)
On my iPhone, under Settings -> Mail, Contacts, etc -> iCloud -> Account,
At the very bottom under "Advanced" is the email info, now this should be the new active @me address but its NOT! It's my old AppleID with the @mac address - which of course no one can use for "real email" anymore.
iCloud on my phone is attempting to use my @mac address for outgoing messages!!!! ARGH!
WORSE is you can not edit this! You can click and see details, change the "name" but you can not change the address!
I think this might be the issue for all of us on this thread - check it out and see. If so it's something we will def. have to have a fix from apple for. :-( That or perhaps delete your iCloud account, your existing Apple ID and get a new @me Apple ID that will work for email as well. (PITA!) I hope we get a fix from Apple.