I haven't yet watched a Facebook live video. Possibly this is because none of my Facebook "friends" has ever made a live video, or perhaps I don't sit on my Facebook news feed enough to notice if one of them did. Periscope, by contrast, shows you live videos on its default screen, and _nothing else_.
A lot of Periscope content is very disposable. The value lies in the live-ness of the videos. Replays can be interesting, but they are secondary. Live video is more "real", because it can't be edited or faked (at least not very easily, and it's impossible to fake live responses to viewers' comments).
If I could choose just one improvement to make to Periscope, it would be to give the user some control over the parameters of the algorithm that determines visibility of new videos on the main list, for example to give priority to specified hashtags. The current algorithm, whatever it is, tends to favour youth over age, probably as a function of the normal human reaction to physical attractiveness (many viewers will continue to watch a new video simply because they like looking at the broadcaster's face, and this will affect the algorithm's decision to highlight that video in the main list for other users of the app).