Comparison of the two Keepass2Android flavors
Keepass2Android 
(https://play.google.com/store/apps/details?id=keepass2android.keepass2android) is a password manager app for Android. It is meant as a companion to KeePass 2 or KeePassXC on PC platforms. Therefore, it is important to provide reliable synchronization between all devices accessing the same password database. For this reason, Keepass2Android comes with extensive support for different protocols and cloud providers, i.e. it can read and write files in many ways (WebDav, SFTP, FTPS, Dropbox, Google Drive, OneDrive, etc.). This is an important feature of Keepass2Android.
However, integrating each of these protocols means that the app needs the Android permission to access the network (or internet). Some users do not like their password manager to have this permission. Because if the app wouldn't have this permission, it couldn't transmit any sensitive content anywhere, even if it was malicious (which it is not).
For those users, the Keepass2Android Offline 
flavor (https://play.google.com/store/apps/details?id=keepass2android.keepass2android_nonet) can serve as a replacement: It can only read/write files through the Android file browser, but therefore does not request Internet permission.
Yes. KP2A (by default) uses a local cache to make databases available even when there is no internet connection. Just because it doesn't have "Offline" in the name, doesn't mean it doesn't work then.
No and Yes. KP2A Offline does not have any built-in support for Web protocols or Cloud providers. This means that the app cannot access a web server or the cloud directly. However, there are two ways to use KP2A Offline, even if you want to use remote storage.
- One way of doing so is to have a locally stored file. Keepass2Android can access this file, and so can a third-party sync app like FolderSync.
- Another way of doing this: Android provides the so called Storage Access Framework which is an abstraction such that a storage provider app can provide files to other apps. It is sufficient if the storage provider has internet permissions (e.g. the OneDrive app). Keepass2Android can "talk" to the storage provider through the Storage Access Framework. Unfortunately, not all cloud providers implement the interface (e.g. Dropbox) or sometimes implement it without write support, and there is limited support for web protocols through Storage Access Framework.
Another issue which appears frequently when using the Storage Access Framework or external apps for synchronization is when it comes to conflict resolution.