MobileMe settings no longer work in my email client?

Those settings are correct. However non-Apple applications are asking for a password because Apple are now requiring app-specific password for third-party applications to access iCloud if you have set up two-step verification (in other words you have to obtain an individual password for any non-Apple program that wants to access iCloud).
Select 'Password and Security'; click 'Generate an App-Specific Password' and follow the instructions. Once you have the password, copy it and paste it into the password field in the application. You should also keep a note of it (though you can generate a new one if required).

If you haven't set up 2-step verification for your Apple ID then it's something else has gone wrong. You don't say what your system is (please always do). If it's a Mac using 10.7.5 or above, uncheck Mail in the list in System Preferences>iCloud and recheck it. If Windows, the same in the iCloud Control Panel. If you are using a Mac earlier than 10.7.5 please see

Entering iCloud email settings manually in Snow Leopard or Leopard

It seems that even Apple's Mail client under Snow Leopard acts like a third party client and now requires an App-Specific Password. I generated and entered an App Specific Password for both incoming and outgoing email, but I'm not sure if two different passwords are required.

That's correct - Mail under Snow Leopard isn't officially intended to access iCloud (and on later systems it's the iCloud preference pane which does the signing in rather than Mail), hence the requirement. AFAIK one password will do for both directions - it's specific to the application rather than the activity.