When comparing Messenger vs WAVE calendar, the Slant community recommends Messenger for most people. In the question“What are the best Material Design apps?” Messenger is ranked 22nd while WAVE calendar is ranked 28th. The most important reason people chose Messenger is:
Allows communicating with anyone who's a friend on Facebook.
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 Available in 10 languages
Available in English, Dutch, Spanish, Swedish, Italian, Portuguese, German, Chinese, and Russian.
Pro Integrates with Google Maps for directions
If a location to an event is added, the app will show a map and allow pulling up Google Maps for directions.
Pro Built-in task manager that can integrates with Google Tasks
Wave allows creating local and Google Tasks to-do items that can be synced across devices and color-coded.
Pro Has day and week, month, list and 2 agenda views
The app has day, week, week-agenda, month, month-agenda and list views.
Day, 3-day and week views split up the days horizontally and hours vertically.
Week agenda splits the screen in 6 equal parts - 5 parts for work days, sixth for weekend. Double-tapping a day will open a list of events for that day. From here swiping left and right allows moving between days.
Month view splits the screen in half with one half displaying a typical month view with color-coded dots next to dates to show which days have events and the other a list of events for the selected day.
Month agenda view lays out all days in a grid and displays all event for the day.
The list view is actually the closest to a typical agenda view despite the name. The difference here is that there are two modes of showing that information - default and detailed. The default mode displays date, time, event name and location (if available) and the detailed mode displays extra information such as what calendar is the event part of, who's attending, notes, etc.
Pro Integrates with Wave Scheduler for event planning
Another app on offer from the publisher is WAVE Scheduler, which integrates into WAVE Calendar for easy event planning.
Pro Color-coding for events and calendars
The app will color-code different calendars in different colors as well as allow selecting a different color for each event.
Pro Includes a store for purchasing various public calendars
The app has a store for sports, stock exchange, holiday, religion and other calendars.
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 Some basic features require signing up
Paid features include the ability to add attachments and travel time to events and see local weather in calendar.
Con Inconsistent design
There's different branding in different places (Still has e-cal branding in some places). Sometimes looking at an event requires tapping twice while most of the time it's once, a horizontal list of items looks differently based on the views it's accessed through, etc.
Con No natural language input support
