Falsificationism can be thought of as an answer to two distinct questions:
How can we respond to Hume’s problem of induction?
How can we distinguish science from non-science, a problem of demarcation?
On the first question, I think that the issue divides into a couple of issues.
i) Can we afford Popper’s solution to the problem of induction? He does not so much answer the problem as sidestep it. The problem of justifying induction is not attempted. Rather, Popper argues that we can make do without induction. But in that case, scientific theories remain, even after severe testing, mere guesses. So how can one rely on any theory. Popper’s reply to this is to deny that we can ever rely on a theory but it is rational to prefer tested theories to untested ones. But why should that be? (If one believes that theories can be confirmed by evidence, then it is obvious why we should prefer successfully tested theories. But Popper does not allow any inductive confirmation. Restricting himself to deduction, Popper relies instead on the idea that a single observation can refute a theory. But a failure to refute a theory is not the same as offering positive reason to prefer it.)
ii) Can the key idea that even a single observation refute a theory (whilst one needs an infnite number to confirm one) really work? The problem is that an observation is not itself something infallibly given. So if an observation and a theory clashes it is not clear which should be rejected. And if not, that makes things much less clear cut than the idea might at first seem.
See also Lakatos' story of planetary misbehaviour. If the movement of a planet appears to conflict with Newtonian physics, that might indicate that Newtonian physics is false. But it might also indicate that something else is affecting the movement although still in accord with Newtonian physics. So that single observation does not refute Newtonian physics. It would only refute it - according to the clear logic of simple falsificationism - if we somehow knew that there was nothing else relevant to the case at hand than Newton's laws and the planetary system already identified. But we don't know that. In general, theories are tested against observations not directly or in isolation but against a background of other theories and assumptions.
On the second main question, the issue remains whether falsificationism can serve as a demarcation criterion if it is not the neat solution to the problem of induction we might have taken it to be. The worry that a theory might simply not be falsifiable under any circumstances looks to be a genuine worry. But perhaps no theory couldn’t be falsified / all theories could be falsified given a suitably relaxed approach. What is less clear – see Lakatos – is that a conflicting observation might ever compel us to regard a theory as falsified. If so the distinction between what is falsifiable and what not might itself look less clear cut. Perhaps no theory has to be rejected given a contrary observation but any theory might be, given a suitably critical scientist.