It sounds like there's a little confusion. Let me elaborate a little bit... I used to be a Senior Advisor for Apple and this was by far the most common call.
First, migrating to a new Apple ID can be difficult, but you can do it. To make sure all future purchases ask for your new Apple ID, go to Settings > iTunes & App Store and then select your Apple ID. Is this your new one or your old one? If it's your old one, tap "Sign Out" and try signing in with your new one. All future purchases should then prompt you for your new Apple ID and password.
However, keep in mind that any apps that you've already downloaded with your old Apple ID will prompt you for that old Apple ID and password when the apps need to be updated. (This is what everyone else is talking about). In order to never see or think about that old Apple ID again, you'll need to delete those apps over time and re-purchase them on your new Apple ID.
Second, it is imperative that you understand what email addresses are and aren't associated with each Apple ID. To check this out, go to id.apple.com and sign in to each of them. It is possible to have multiple emails associated with an Apple ID. It is also possible to have an "alternate Apple ID," which basically means you can use either to sign in.
One more thing—with the announcement of iOS 8, Apple IDs can be conjoined or paired using Family Sharing. It may just be more confusing, but check out the synopsis on Apple's website here.
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TL;DR: keep the one that looks like a bag, it now a universal app. The cart one was the iPad specific one.
Looks like the one of the left (with the shopping cart icon) is the old one. Launching it now directs me to the new one, but it didn't when I was trying to figure this out before. Here's what I get when launching the old one now.
EDIT-20140917: It appears that Apple may have had a separate iPad app that they have since deprecated and made their iPhone/iPod Touch app a universal app and that's why this app is now defunct.