When comparing Messenger vs Threema, the Slant community recommends Threema for most people. In the question“What are the best messaging apps for Android?” Threema is ranked 6th while Messenger is ranked 21st. The most important reason people chose Threema is:
Registration is handled by random generated IDs. No phone number needed.
Specs
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Pros

Pro Uses Facebook friends to create a contacts list
Allows communicating with anyone who's a friend on Facebook.
Pro Read receipts
Can see who has seen each message in group messages.
Pro Easy to share posts from Facebook
Integration with Facebook makes it easy to share any interesting posts from the Facebook timeline.
Pro Great UI
User interface of the application is clean, intuitive and easy to overview.
Pro Registration without phone number
Registration is handled by random generated IDs. No phone number needed.
Pro Encrypted communication
Provides complete end-to-end encryption.
Pro Company behind Threema is based in Switzerland
Pro Servers in Switzerland
Privacy laws protect the data Threema has.
Pro Supports encrypted group chats
Pro Provides a web client
You can control and message with friends via Threema Web, the official web client for Threema.
Pro Ability to send pics and videos
Threema supports sending media messages such as video, pics and even voice messages.
Pro Group chat polls
Pro Allows verification of contacts identity via QR Code
Cons
Con Lack of privacy
Messenger is owned by Facebook. Based on their privacy policy, this means that any information you provide (including messages sent) are used by Facebook to profile you and sell access to your data to advertisers.
Con Lack of end-to-end encryption
Secret (end-to-end encrypted) chats are only supported in one-to-one messages, and are not enabled by default, and the encryption has not been verified independently.
Con Not free
Unlike many other options Threema is not free as it costs $1.99 for the app.
Con Single device only
You cannot use multiple devices to send messages. However, it does have message mirroring on a web client.
Con No cloud sync
E2E encryption means no cloud syncing of messages.
