When comparing GroupMe vs Telegram, the Slant community recommends Telegram for most people. In the question“What is the best team chat software?” Telegram is ranked 4th while GroupMe is ranked 51st. The most important reason people chose Telegram is:
Copy/paste videos, gifs, images, geolocation, documents, and even executable files. Drag over files for practically any time up to 1.5GB in size. Images can be compressed, too.
Specs
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Pros

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.
Pro Free and convenient large file sharing
Copy/paste videos, gifs, images, geolocation, documents, and even executable files. Drag over files for practically any time up to 1.5GB in size. Images can be compressed, too.
Pro Privacy focused
In addition to the end-to-end encryption that provides privacy to communications, the app also includes self destructing timers for chats, and the ability to permanently delete messages, yours and theirs, from your device.
Pro Free of cost
The application is available free of cost.
Pro Media messaging
Videos, gifs, images, geolocation, and documents.
Pro Beautiful UI
Very customizable with a lot of clean looking options.
Pro Channels for one way broadcasting
Broadcasting channels suitable for announcing things.
Pro Supergroups of up to 200,000 people
In Supergroups, anyone along with admins can delete their messages from all sides. A message can be pinned to the top. All of the user's messages can be deleted by an administrator, and they can be banned and reported as well. Supergroups can have a public link (outside of Asia, due to high spam volumes), which is also a username.
Pro Two-step verification
You can set up an additional password to your account, which can be reset by your email. You don't have to use an email if you do not want to. If you lose access to your email or forget your password, you will be able to reset it via text message if your account is over seven days old and will receive it within a week (in case you recover anything before then).
Pro Possible to send photos and videos without compression
Pro Secure
Pro Save time with inline bots and link previews
Unfortunately, on the desktop videos don't play, but images are viewable without visiting popular websites like imgur and Instagram.
You can play YouTube videos and others within the app, so you don't need to embed code or anything.
With inline bots, such as @gif, @pic/bing, @vid/YouTube, etc, you can look things up without leaving the client or chat that you're in. Simply type the name and some keywords and wait for the results to load.
Pro High quality calls
Uses the latest GPL audio codecs, such as Opus, an adaptative high-quality codec that supersedes AAC at any bitrate.
Pro Desktop app handles HIDPI
The desktop app has a built in setting to scale the UI of the app up to 200% which allows for those with 4k monitors/screens to use the app without scaling issues.
Pro Cloud chat keys are kept in different jurisdiction
Even though encryption used for cloud-based storage (e.g. Gmail, Facebook, iCloud) may not be as secure as end-to-end encryption or zero-knowledge storage, Telegram does the most that it can by holding keys to chats in different jurisdictions, requiring intergovernmental cooperation to obtain the keys. This is aside from whatever other security measures have been taken, of course.
Pro Saved messages
You can write messages to yourself. You can write some notes, add links, and see everything across all your devices.
Pro Customer support is within the app
Although it's volunteer based, but still pretty top-notch if available, which it tends to be.
Pro Voice and video messages
You can post voice and video messages if you are too lazy to write text messages.
Pro @mentions
You can ping people to get their attention by @mentioning them.
Pro Open source client apps
Client apps are open source, making whole service more trustworthy.
Pro Multiplatform native client apps
There are native desktop, mobile and web client apps, that are optimized for speed and are light on memory usage.
Pro Pinned Messages
You can pin messages to the top of groups or supergroups so members will see them when opening the group.
Pro Unlimited storage
Unlimited number of files can be stored forever on the cloud. You can upload files up to 1.5 GB each.
Pro Popular in the USA
On SimilarWeb you can see that Telegram is on the top 15 among all communication apps.
Pro Banned in China
Pro Self-destruct messages
Messages can be set on a timer. They will disappear after the timer expires.
Cons
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.
Con Not fully open source
Telegram has an open API and protocol free for everyone, but the backend software is not open source. Therefore 100% privacy can not be evaluated.
Con Can only be used with a phone number
It is not possible to activate and use Telegram by mail etc. You need a phone number.
Con Only "Secret Chat" messages are encrypted end to end
The only way to get end-to-end encryption (client-client) using Telegram is to use the Secret Chat function, anything through the service is server-client encrypted. Also secret chat is only implemented in the mobile apps, meaning there is no way to encrypt messages end-to-end when using the desktop client or the web browser client.
Con "Secret Chat" feature is not available for desktop client
Secret chats with end to end encryption are only available on mobile apps.
Con Does not support text messaging
You can NOT make it your default text messaging App.
Con Does not work for visually impaired
Doesn't work for visually impaired.
Con No source code sintax highlighting
Telegram does not have source code syntax highlighting, making it difficult to share code snippets.
Con Does not support spherical 360° photos
Spherical 360° photos becoming more popular these days, but Telegram does not support such format.