When comparing Threema vs GroupMe, the Slant community recommends Threema for most people. In the question“What are the best messaging apps for Android?” Threema is ranked 6th while GroupMe is ranked 23rd. 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 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

Pro Reads receipts
GroupMe has a built in function to read receipts.

Pro Great web client
GroupMe has an excellent web client that can be used to send and receive messages and manage groups from the desktop.

Pro IFTTT Integration
Allows sending messages from web services via IFTTT.

Pro Integrated calendar
It's possible to add location and time with reminders directly in the GroupMe app.

Pro Makes it easy to silence 1 or all groups
GroupMe has settings that allow muting a single group or turn off all GroupMe notifications from within the app.

Pro Does not require everyone in the group to have a smartphone
GroupMe can set up a custom phone number which allows people without the App to send and receive messages. This makes it easy to add anyone to messaging.
Cons
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.
Con Drains battery fast
GroupMe, in order to have a constant connection which allows for notifications drains the battery of a device quickly, which is not ideal for a messaging app that one needs on time notifications from.
