After verifying my billing information, iPhone still won't let me download apps from the App Store

Re: "Why won’t my phone let me down load apps : asks me to put in my password to verify my billing information

and when I do that, it still won’t download the app"

As Apple users, like yourself, in the community, we have no access to your accounts or details)

There are various reasons that it could be, so you might want to check through some of the possibilities:

You can check you are correctly signed in (to your iPhone) with your Apple ID:

Sign in with your Apple ID - Apple Support

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You might want to log-in to your Apple ID account online and check all your details are correct and up-to-date:

(You can add to, or edit your details, (eg: card expiry date etc.) if you wish.) 

https://appleid.apple.com

If needed: How to: Get a verification code and sign in with two-factor authentication - Apple Support

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You can check whether there are restrictions in place in ScreenTime:

(Check: Settings > ScreenTime > Content and Privacy Restrictions > iTunes & App Store Purchases)

and

How to: Manage App Store purchases, subscriptions, settings, and restrictions on iPhone - Apple Support

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If you have disputed a charge from Apple:

you may need to ask your card provider to allow payments to Apple again:

Also see: What to do: If you can’t download or update apps - Apple Support

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If needed (eg: info. / support / contact options etc.):

App Store Support website App Store - Official Apple Support

and

Billing and Subscriptions Support website: Subscriptions and Billing - Official Apple Support