Just like Meerkat, Periscope allows you to broadcast what you're doing through live video. Where Meerkat feels utilitarian, though, Periscope appears to have been more carefully designed. Arguably, that should be the case: as The Verge points out, despite the fact that Twitter acquired Periscope in January, the app has been in development for over a year.
The result is an app that looks like it actually... does stuff. Perhaps the biggest obvious advantage on display is that Periscope allows users to save streams on the service—in fact, it does so by default—so that others can replay them later. If you've found Meerkat frustrating because you don't see streams in time, Periscope solves your problem.
Elsewhere, there's also more interaction available than Meerkat offers: you can send hearts to a person streaming on Periscope to let them know that you like what you see. Hearts appears on-screen for all other streamers to see, so they serve as a real-time indicator of popularity that's perhaps more reliable than the number of people of watching. You can also send hearts more than once, presumably to let the streamer know that you still dig what they're showing.
Aside from notifications from other Periscope users that you choose to follow, that's about it. For now, Periscope isn't baked directly into Twitter; it's its own app—though it's not clear how long that will last. If you've an iPhone you can give it a try now. An Android version is apparently in the works. [Periscope via Verge]