Einstein is being much more profound and subtle than Eddington, but even Eddington's quote has been misunderstood. Eddington is not saying that the Second Law of Thermodynamics can never be violated, in fact, in other passages of his, which I have posted on other places on this site, he explicitly says it is possible. No. What Eddington is saying is that if your theory qua theory proves that as a rule entropy need not increase, then there is no hope for you. In very exceptional or rare cases which, btw, won't be reproducible by other experimenters since they are freakish, Eddington admits the Second Law can be Violated.

Einstein is being more interesting. Note well his caginess in qualifying his statement: "within the framework of the applicability of its basic concepts." Obviously he means at least "assuming it is within the realm of validity of the approximations it makes", e.g., we have to be talking about macroscopic bodies, macroscopic properties, and macroscopic variables. Not Szilard engines. But he means more, and in two interesting ways.

A. As anna v. points out in a comment somewhere, Thermodynamics or Stat Mech are mathematical theories which make certain idealisations from the real world. I rarely agree with the exact way she phrases things, but here Max Planck in his very old book on Thermodynamics makes explicit one of the idealisations made by Thermodynamics, so that its results are only approximations: we always assume that the dynamical system passes from its current state to the most probable macro-state which it can pass to. The Laws of Thermodynamics, indeed, its entire mathematical structure, rest on this assumption, according to Planck. So Einstein is saying that provided this assumption holds true, the conclusions of Thermodynamics will never be overthrown. And Einstein was certain that the conclusions of Quantum Mechanics would be overthrown, even within its own realm, by a more complete deterministic theory with hidden variables. So he is implicitly contrasting Thermodynamics with that and making Thermodynamics more fundamental than Quantum Mechanics. This is saying a lot more than just that the experimental observations which, to date, support Thermodynamics, will never be contradicted in the future.

B. Einstein, I feel sure (I guess that is a kind of disclaimer...the more someone uses words like "sure", the less they ....) was also contrasting Thermodynamics with, e.g., the Newtonian theory of gravity. People are fond of saying that the Newtonian theory retains its validity as a good approximation. But Einstein is asserting that Thermodynamics is much more than just a good approximation. Newtonian gravity is a bad approximation to events near black holes, and even the principle of the constancy of the speed of light (Special Relativity) is a bad approximation near a black hole. Einstein is asserting that the logical structure of Thermodynamics will be retained in all future physical theories, no matter what new forces are discovered, and that it can and will be erected on top of whatever new fundamental physics comes along, substantially unchanged. The new fundamental physics will find the new places where the old fundamental physics is a bad approximation, but Thermodynamics will remain valid even in those new places.

These are not necessarily my own opinions.

Einstein is being much more profound and subtle than Eddington, but even Eddington's quote has been misunderstood. Eddington is not saying that the Second Law of Thermodynamics can never be violated, in fact, in other passages of his, which I have posted on other places on this site, he explicitly says it is possible. No. What Eddington is saying is that if your theory qua theory proves that as a rule entropy need not increase, then there is no hope for you. In very exceptional or rare cases which, btw, won't be reproducible by other experimenters since they are freakish, Eddington admits the Second Law can be Violated.

Einstein is being more interesting. Note well his caginess in qualifying his statement: "within the framework of the applicability of its basic concepts." Obviously he means at least "assuming it is within the realm of validity of the approximations it makes", e.g., we have to be talking about macroscopic bodies, macroscopic properties, and macroscopic variables. Not Szilard engines. But he means more, and in two interesting ways.

A. As anna v. points out in a comment somewhere, Thermodynamics or Stat Mech are mathematical theories which make certain idealisations from the real world. I rarely agree with the exact way she phrases things, but here Max Planck in his very old book on Thermodynamics makes explicit one of the idealisations made by Thermodynamics, so that its results are only approximations: we always assume that the dynamical system passes from its current state to the most probable macro-state which it can pass to. The Laws of Thermodynamics, indeed, its entire mathematical structure, rest on this assumption, according to Planck. So Einstein is saying that provided this assumption holds true, the conclusions of Thermodynamics will never be overthrown. And Einstein was certain that the conclusions of Quantum Mechanics would be overthrown, even within its own realm, by a more complete deterministic theory with hidden variables. So he is implicitly contrasting Thermodynamics with that and making Thermodynamics more fundamental than Quantum Mechanics. This is saying a lot more than just that the experimental observations which, to date, support Thermodynamics, will never be contradicted in the future.

B. Einstein, I feel sure (I guess that is a kind of disclaimer...the more someone uses words like "sure", the less they ....) was also contrasting Thermodynamics with, e.g., the Newtonian theory of gravity. People are fond of saying that the Newtonian theory retains its validity as a good approximation. But Einstein is asserting that Thermodynamics is much more than just a good approximation. Newtonian gravity is a bad approximation to events near black holes, and even the principle of the constancy of the speed of light (Special Relativity) is a bad approximation near a black hole. Einstein is asserting that the logical structure of Thermodynamics will be retained in all future physical theories, no matter what new forces are discovered, and that it can and will be erected on top of whatever new fundamental physics comes along, substantially unchanged. The new fundamental physics will find the new places where the old fundamental physics is a bad approximation, but Thermodynamics will remain valid even in those new places. For example, it remains valid for black holes.

These are not necessarily my own opinions.