I'm rewatching now and one question has just ocurred to me. Why are the companions and the Doctor only starting to see the Silence now? What has changed? I mean, I get that they might have seen them before and then forgot about them, but why start keeping score now, why have they never been a problem before? I know that the writers only imagined them much later but I would like to know if there's an in-universe explanation.
but why start keeping score now, why have they never been a problem before?
They might have been, and everyone just forgot about them again. Remember that Amy is able to take a picture by chance. The aliens weren't aware that this kind of technology exists in 1960's Earth (which originally wasn't the case obviously).
Remember that the Doctor doesn't mind aliens living on Earth as long as they don't cause trouble. Considering what he said in those two episodes, he didn't know the Silence would drive humankind to the stars, which wouldn't be bad on its own. Also he got on their trail when looking for the scared child. If you scare a child in front of the Doctor, you're typically in trouble, even if it's just a very long monologue.
My presumption was that the events of the previous series, culminating in "The Big Bang" had made major changes to the history of the cosmos. There were massive internal retcons, if you will.
According to this hypothesis, prior to previous series's events, the timeline had not even contained the Silence. So there was nothing to notice on earlier trips to Earth. Now, however, when the Doctor arrives on contemporary Earth, after the changes have been made, the Silence gets noticed in pretty short order. (The silence wasn't ready to hide itself from observers coming from outside the normal time stream.)
Remember that in The Time of the Doctor, the Doctor identifies the Silence to Clara as the confessors of the Papal Mainframe. He might have learned what they were since The Impossible Astronaut, or he might have known about them the entire time.
The reason they start looking for the Silence and counting them is because they killed the random bystander in front of Amy, and told her to tell the Doctor "what he must know and what he must never know." After they encounter the Silence hive in Florida, they must have been able to see one long enough to start taking notes. Or something that leads to them tracking down the Silence.
So before that, they didn't have a reason to remember or fight. Any encounter with the Silence were presumably not the splinter cell, and thus not a problem.
Of course, the real reason is that they hadn't been invented yet. The same reason we never saw Weeping Angels before Blink, etc.