"Falsifiable" is a fantastic term but the philosophy of science did not stop with Popper, and it's not the only term we have available to discuss competing theories.
"More accurate" is a good way to phrase it. "More accurate" also describes GR when you compare it to Newtonian mechanics. Strictly speaking, Newtonian mechanics is false and has been falsified if you subscribe to the Popperian view. However, if Newtonion mechanics is false, is it useless? No.
As it turns out, there are a few problems with the epistemology of falsifiability. The main ones I can think of are:
1. Duhem-Quine problem - there is a large number of auxiliary hypotheses under test in any experiment, in addition to the primary hypotheses. These very numerous and we need a framework for deciding how to apply the results of our experiment to the many hypotheses.
2. Statistical claims may be unfalsifiable. Consider a theory that claims that a coin flip has a 50% probability of being heads... how does one falsify this? One can't strictly falsify it, but you can show that the evidence is unlikely given the hypothesis. So we need some framework that connects statistical and probabilistic evidence to our knowledge of the world.
Or in summary, the problems with falsifiability are that falsifying a theory doesn't give us the information that we want, and it's impossible to falsify many theories. To abuse analogies, falsifiability is kind of like trying to cross a river with your car, when there's no bridge and the car won't start.
One approach other than Popperian falsifiability is a Bayesian system of belief and likelihood. This only one direction that the philosophy of science is exploring, but it is probably the one most familiar to HN readers.
I don't have resources for Bayesian systems of belief. I do like Error and the Growth of Experimental Knowledge by Deborah Mayo, but the book is a bit intimidating. I admit I haven’t finished reading it, either.