If the close connections involve changing onto a Regional train, then it doesn't matter if you miss the expected train out because you can take the next train. A Regional ticket from Florence to Manarola is OK on any Regional trains on the route for six hours after validation (time/date-stamping) before boarding, and one from la Spezia to Manarola for 75 minutes.
It is only worth buying in advance if for some reason you really want to take a connection which includes a Frecciarossa or Intercity train with reserved seats between Pisa and la Spezia. And at most times of day, these connections aren't worth the extra money, as they commit you to the train you have reserved seats on. And if you miss that train in Pisa Centrale because of a late train from Florence, you can only change your ticket onto the next train to la Spezia of a similar class, you can't use it on a Regional train.
I'd buy a Regional ticket to Manarola in Florence, as missing connections is then no problem at all.
Edited: 3:09 pm, August 18, 2013