Any apps your purchased with the forgotten username and password can not be downloaded and/or updated with the new username and password. If that matters.
If you've purchased anything that you care about it would be worth your while to contact Apple Support and try to retrieve this Apple ID. Even if you have not, remember this ID has your credit card info, it would be worth while to get it back so you can change that info, or remove it. Just as a precaution.
If you still have not purchased anything with the new ID (but have with the old), you should consider abandoning the new one instead. It would make your life simpler.
That said, to get things working with the new ID on the iPad and in iTunes.
On the iPad, make sure you go into iTunes, the App Store, and iBooks store (if you have it) got to the bottom left of the main screen. If it shows the right Apple ID you should be OK, if not tap to sign out, then tap again o sign in with the correct ID.
In iTunes on the computer, do the same thing in the top right corner where it should either be showing your Apple ID or a log in (sign on) option. If you are not signed in with the new Apple ID, do so.
Under the Store menu be sure to authorize your computer.
Also, under Settings > General > Restriction, make sure you did not turn off the ability to get new apps. I should have probably mentioned that first.
Good luck