The major difference is that an individual reddit account could only press the button once and anonymous presses were not allowed.
A single person can send an unlimited number of messages to this new website.
Given that I would expect this to go longer.
On the other hand, The Button was an april fools experiment on one of the internet's most popular websites and thus garnered a ton of attention. This could easily fade into obscurity much faster.
Ah, here we go:
https://en.wikipedia.org/wiki/The_Button_(Reddit)
Looks like it lasted about 4 months. So I guess that gives us a baseline for the over-under on the lifetime of this site.