Everyday.me is a timeline for your eyes only.
Everyday.me for the iPhone is like a Facebook timeline for you, and you alone. You get to generate a massive combined timeline of all your Tweets and Facebook statuses in the past. It's very similar to Momento, which I have been using for the past year. Just like Momento, you can write entries to be viewed in the app, or import your existing social media feeds. For now, the app only supports Twitter, Facebook and Instagram.
It seems easier to glance through your feeds in Everyday.me because you can scroll through it endlessly back in time, but there doesn't seem to be a way to go to a specific date like in Momento.
Another difference is the ability to view your feed online in a web browser, which also means your feed is backed up in the cloud. Momento does not do this, but it allows you to back up your data through iTunes file transfer. Another interesting feature is the quaterly/annual reports, where you receive a summary of what you've been up to all these months.
It's not yet available on the AppStore, but you can sign up for a private beta at Everyday.me.