I deleted Facebook several years ago, but that's not where I'm going with this answer, although you will get people who don't have or want a Facebook.
Facebook is a poor way to share persistent information. The newsfeed rapidly scrolls past, and comment threading is rather poor, and not built for information-dense sharing like you might get when working through a tricky problem. When you post "Abstract deadline next Friday!" it will disappear within days, potentially not being shown to people who log on a few days later.
Furthermore, if this page is for the entire institute (I assume you mean university), you are going to get many posts from every discipline. I don't need my page full of physics seminars and pharmacology retreats.
For this, I would suggest Slack (or, hesitantly, Discord). For one, Slack already has an air of professionalism, and many groups I know use it internally.
But with Slack/Discord, you can create channels for: e.g. announcements, deadlines, events, software help, writing help, and memes/off-topic. That way each topic is canalized, and if I don't want to help people write, I can just skip that channel, and I can totally ignore the ones not relevant to my field.
I hesitate to recommend Discord because it is very confusing for non-technical people; even more so than Slack, IMO. And, as of right now, it will be mostly gamers who have it installed. Many other types of people will have Slack, and it has decent integrations with cloud services like Google and Microsoft. I suspect you would have more difficulty setting up Slack in some fields compared to others. In that sense, Facebook is a good choice for ease of accessibility.
That said, Discord does have some powerful tools to, for example, tag someone as a physicist, which allows them to see the physics channels and nothing else. Discord is also seen as much more casual; but others on this site have posted they've had a lot of success with it (but when their field is on their profile, they're all in computer science!).
I said in a comment, calendar integration is also critical. I and many others live off their Google Calendar or similar. So if you have a nice calendar system, where I can sign up for an event (then you can see how many people are gonna come), and sync it to my calendar (but not the other, irrelevant events), I would use your system much more. Discord has some decent tools for this, not sure about Slack.