Twitter this week released a new update for iOS, for "bug fixes" but it seems that the new version came with a bug.
Device owners with iOS they report to iTunes that the 6.1.1 version of Twitter came up with a bug that prevents the application from starting. It is not clear if this problem affects all or just some devices that have been "tampered with". However, Twitter should release the new version soon.
If you have already installed the latest version, unfortunately, there is no way to "download" it to the previous one, unless you have the old IPA in your iTunes library or perhaps in a recent backup.