Hey Community,
to answer directly the topic question: Yes, they do.
But to answer directly a concern: It is not on the security/encryption side.
There are a lot of forum posts where it is at first unclear wether the people talk about the mobile applications or the ones for the desktop. But to actually provide help this is a crucial fact. Also there exist many questions why does this work on the desktop and not my smartphone. This thread wants to summarize the differences so that you directly know how Cryptomator works on your platform.
What is their difference?
Apart from the obvious OS question, it is how they access your encrypted files and making the decrypted versions accessible.
Desktop
When you unlock a vault in the Cryptomator desktop application and browse through its content, you do it with the system file browser. This means that every request to read, write or move a file/directory is generated by the OS, then channeled through a “Cryptomator layer” and finally executed by the OS again.
Therefore the general rule of thumb is that the vault must be accessible via a system path (e.g. /home/jim/vault/) of your operating system_. There are exceptions, thou. But if your vault is stored on your hard drive, it definitely works. The OS capabilities to access remote files can be extended by drivers or third party applications, e.g. MountainDuck or davfs, but it might introduce incompabilities.
Android
On Android your vault is not stored on your smartphone or tablet. Cryptomator connects with your cloud provider, queries the encrypted directories and shows a locally decrypted view. Also the content of a file is only downloaded on demand, i.e. when you want to view or edit it.
iOS
Also here, Cryptomator connects directly to the cloud and sends requests to download this file or list that encrypted directory of the vault and provides on your device a locally decrypted view. The difference to Android is two fold: First, similar to the desktop versions, the app integrates into the system, so you can work on it like any directory in the system browser. And second, once a file is downloaded, it is stored on the device until the user manually deletes the cache.