(1) Rule of law is an intrinsically moral notion. Indeed, I don't see how one can have a consistent theory of rule of law without appealing either to natural law theory or to some higher rule by law (e.g., divine command theory).
(2) Rule by law is very different, despite some superficial similarities. Rule by law is prudential: one rules by law (properly speaking) not because the law is higher than oneself but because it is convenient to do so and inconvenient not to do so. In rule of law, the law is something the government serves; in rule by law, the government uses law as the most convenient way to govern.
(3) The two chief arguments for rule by law rather than rule of law are exactly the same ones that are always used against natural law theory:
(a) disagreement and uncertainty in moral judgments;
(b) the claim that rule of law is seminal anarchy.
(4) The chief arguments against rule by law and for rule of law are exactly the same ones that are always used against the opponents of natural law theory:
(a) the question of how one can have authority without any moral basis;
(b) the claim that rule by law is seminal despotism.
(5) Rule by law can be either ad hoc (which is genuine despotism) or principled. Principled rule by law theory shares with rule of law theory the arguments that a stable, generally recognized law is needed in order to maintain generality, impersonality, and effectiveness of government. Thus principled rule by law theory allows for what Fuller has called "the internal morality of law" to the extent that this is prudentially justifiable as conducive to the ends of government. (There is an interesting paper by Kenneth Winston on this subject in the context of Chinese Legalism at SSRN; much of what I say in this post is influenced by Winston.)
(6) Much of what we call rule of law today is really rule by law; a very serious equivocation given that they tend in entirely different directions.
UPDATE: corrected some rather significant typographical errors.